modelparameters.sympy.functions.special package

Submodules

modelparameters.sympy.functions.special.bessel module

class modelparameters.sympy.functions.special.bessel.AiryBase(*args)

Bases: Function

Abstract base class for Airy functions.

This class is meant to reduce code duplication.

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.BesselBase(nu, z)

Bases: Function

Abstract base class for bessel-type functions.

This class is meant to reduce code duplication. All Bessel type functions can 1) be differentiated, and the derivatives expressed in terms of similar functions and 2) be rewritten in terms of other bessel-type functions.

Here “bessel-type functions” are assumed to have one complex parameter.

To use this base class, define class attributes _a and _b such that 2*F_n' = -_a*F_{n+1} + b*F_{n-1}.

property argument

The argument of the bessel-type function.

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

property order

The order of the bessel-type function.

class modelparameters.sympy.functions.special.bessel.SphericalBesselBase(nu, z)

Bases: BesselBase

Base class for spherical Bessel functions.

These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order.

To use this class, define the _rewrite and _expand methods.

default_assumptions = {}

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.bessel.SphericalHankelBase(nu, z)

Bases: SphericalBesselBase

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.airyai(arg)

Bases: AiryBase

The Airy function operatorname{Ai} of the first kind.

The Airy function operatorname{Ai}(z) is defined to be the function satisfying Airy’s differential equation

\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]

Equivalently, for real z

\[\operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]

Examples

Create an Airy function object:

>>> from ... import airyai
>>> from ...abc import z

>>> airyai(z)
airyai(z)

Several special values are known:

>>> airyai(0)
3**(1/3)/(3*gamma(2/3))
>>> from ... import oo
>>> airyai(oo)
0
>>> airyai(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(airyai(z))
airyai(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(airyai(z), z)
airyaiprime(z)
>>> diff(airyai(z), z, 2)
z*airyai(z)

Series expansion is also supported:

>>> from ... import series
>>> series(airyai(z), z, 0, 3)
3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airyai(-2).evalf(50)
0.22740742820168557599192443603787379946077222541710

Rewrite Ai(z) in terms of hypergeometric functions:

>>> from ... import hyper
>>> airyai(z).rewrite(hyper)
-3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

See also

airybi

Airy function of the second kind.

airyaiprime

Derivative of the Airy function of the first kind.

airybiprime

Derivative of the Airy function of the second kind.

References

[1]

http://en.wikipedia.org/wiki/Airy_function

[2]

http://dlmf.nist.gov/9

[3]

http://www.encyclopediaofmath.org/index.php/Airy_functions

[4]

http://mathworld.wolfram.com/AiryFunctions.html

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

nargs = {1}

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

unbranched = True

class modelparameters.sympy.functions.special.bessel.airyaiprime(arg)

Bases: AiryBase

The derivative operatorname{Ai}^prime of the Airy function of the first kind.

The Airy function operatorname{Ai}^prime(z) is defined to be the function

\[\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.\]

Examples

Create an Airy function object:

>>> from ... import airyaiprime
>>> from ...abc import z

>>> airyaiprime(z)
airyaiprime(z)

Several special values are known:

>>> airyaiprime(0)
-3**(2/3)/(3*gamma(1/3))
>>> from ... import oo
>>> airyaiprime(oo)
0

The Airy function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(airyaiprime(z))
airyaiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(airyaiprime(z), z)
z*airyai(z)
>>> diff(airyaiprime(z), z, 2)
z*airyaiprime(z) + airyai(z)

Series expansion is also supported:

>>> from ... import series
>>> series(airyaiprime(z), z, 0, 3)
-3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airyaiprime(-2).evalf(50)
0.61825902074169104140626429133247528291577794512415

Rewrite Ai’(z) in terms of hypergeometric functions:

>>> from ... import hyper
>>> airyaiprime(z).rewrite(hyper)
3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))

See also

airyai

Airy function of the first kind.

airybi

Airy function of the second kind.

airybiprime

Derivative of the Airy function of the second kind.

References

[1]

http://en.wikipedia.org/wiki/Airy_function

[2]

http://dlmf.nist.gov/9

[3]

http://www.encyclopediaofmath.org/index.php/Airy_functions

[4]

http://mathworld.wolfram.com/AiryFunctions.html

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

nargs = {1}

unbranched = True

class modelparameters.sympy.functions.special.bessel.airybi(arg)

Bases: AiryBase

The Airy function operatorname{Bi} of the second kind.

The Airy function operatorname{Bi}(z) is defined to be the function satisfying Airy’s differential equation

\[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\]

Equivalently, for real z

\[\operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.\]

Examples

Create an Airy function object:

>>> from ... import airybi
>>> from ...abc import z

>>> airybi(z)
airybi(z)

Several special values are known:

>>> airybi(0)
3**(5/6)/(3*gamma(2/3))
>>> from ... import oo
>>> airybi(oo)
oo
>>> airybi(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(airybi(z))
airybi(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(airybi(z), z)
airybiprime(z)
>>> diff(airybi(z), z, 2)
z*airybi(z)

Series expansion is also supported:

>>> from ... import series
>>> series(airybi(z), z, 0, 3)
3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airybi(-2).evalf(50)
-0.41230258795639848808323405461146104203453483447240

Rewrite Bi(z) in terms of hypergeometric functions:

>>> from ... import hyper
>>> airybi(z).rewrite(hyper)
3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

See also

airyai

Airy function of the first kind.

airyaiprime

Derivative of the Airy function of the first kind.

airybiprime

Derivative of the Airy function of the second kind.

References

[1]

http://en.wikipedia.org/wiki/Airy_function

[2]

http://dlmf.nist.gov/9

[3]

http://www.encyclopediaofmath.org/index.php/Airy_functions

[4]

http://mathworld.wolfram.com/AiryFunctions.html

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

nargs = {1}

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

unbranched = True

class modelparameters.sympy.functions.special.bessel.airybiprime(arg)

Bases: AiryBase

The derivative operatorname{Bi}^prime of the Airy function of the first kind.

The Airy function operatorname{Bi}^prime(z) is defined to be the function

\[\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.\]

Examples

Create an Airy function object:

>>> from ... import airybiprime
>>> from ...abc import z

>>> airybiprime(z)
airybiprime(z)

Several special values are known:

>>> airybiprime(0)
3**(1/6)/gamma(1/3)
>>> from ... import oo
>>> airybiprime(oo)
oo
>>> airybiprime(-oo)
0

The Airy function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(airybiprime(z))
airybiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(airybiprime(z), z)
z*airybi(z)
>>> diff(airybiprime(z), z, 2)
z*airybiprime(z) + airybi(z)

Series expansion is also supported:

>>> from ... import series
>>> series(airybiprime(z), z, 0, 3)
3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane:

>>> airybiprime(-2).evalf(50)
0.27879516692116952268509756941098324140300059345163

Rewrite Bi’(z) in terms of hypergeometric functions:

>>> from ... import hyper
>>> airybiprime(z).rewrite(hyper)
3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)

See also

airyai

Airy function of the first kind.

airybi

Airy function of the second kind.

airyaiprime

Derivative of the Airy function of the first kind.

References

[1]

http://en.wikipedia.org/wiki/Airy_function

[2]

http://dlmf.nist.gov/9

[3]

http://www.encyclopediaofmath.org/index.php/Airy_functions

[4]

http://mathworld.wolfram.com/AiryFunctions.html

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

nargs = {1}

unbranched = True

modelparameters.sympy.functions.special.bessel.assume_integer_order(fn)

class modelparameters.sympy.functions.special.bessel.besseli(nu, z)

Bases: BesselBase

Modified Bessel function of the first kind.

The Bessel I function is a solution to the modified Bessel equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\]

It can be defined as

\[I_\nu(z) = i^{-\nu} J_\nu(iz),\]

where \(J_\nu(z)\) is the Bessel function of the first kind.

Examples

>>> from ... import besseli
>>> from ...abc import z, n
>>> besseli(n, z).diff(z)
besseli(n - 1, z)/2 + besseli(n + 1, z)/2

See also

besselj, bessely, besselk

References

[1]

http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.bessel.besselj(nu, z)

Bases: BesselBase

Bessel function of the first kind.

The Bessel J function of order nu is defined to be the function satisfying Bessel’s differential equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,\]

with Laurent expansion

\[J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),\]

if \(\nu\) is not a negative integer. If \(\nu=-n \in \mathbb{Z}_{<0}\) is a negative integer, then the definition is

\[J_{-n}(z) = (-1)^n J_n(z).\]

Examples

Create a Bessel function object:

>>> from ... import besselj, jn
>>> from ...abc import z, n
>>> b = besselj(n, z)

Differentiate it:

>>> b.diff(z)
besselj(n - 1, z)/2 - besselj(n + 1, z)/2

Rewrite in terms of spherical Bessel functions:

>>> b.rewrite(jn)
sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)

Access the parameter and argument:

>>> b.order
n
>>> b.argument
z

See also

bessely, besseli, besselk

References

[1]

Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 9”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

[2]

Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1

[3]

http://en.wikipedia.org/wiki/Bessel_function

[4]

http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.bessel.besselk(nu, z)

Bases: BesselBase

Modified Bessel function of the second kind.

The Bessel K function of order \(\nu\) is defined as

\[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},\]

where \(I_\mu(z)\) is the modified Bessel function of the first kind.

It is a solution of the modified Bessel equation, and linearly independent from \(Y_\nu\).

Examples

>>> from ... import besselk
>>> from ...abc import z, n
>>> besselk(n, z).diff(z)
-besselk(n - 1, z)/2 - besselk(n + 1, z)/2

See also

besselj, besseli, bessely

References

[1]

http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.bessel.bessely(nu, z)

Bases: BesselBase

Bessel function of the second kind.

The Bessel Y function of order nu is defined as

\[Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)},\]

where \(J_\mu(z)\) is the Bessel function of the first kind.

It is a solution to Bessel’s equation, and linearly independent from \(J_\nu\).

Examples

>>> from ... import bessely, yn
>>> from ...abc import z, n
>>> b = bessely(n, z)
>>> b.diff(z)
bessely(n - 1, z)/2 - bessely(n + 1, z)/2
>>> b.rewrite(yn)
sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)

See also

besselj, besseli, besselk

References

[1]

http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.bessel.hankel1(nu, z)

Bases: BesselBase

Hankel function of the first kind.

This function is defined as

\[H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.

It is a solution to Bessel’s equation.

Examples

>>> from ... import hankel1
>>> from ...abc import z, n
>>> hankel1(n, z).diff(z)
hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2

See also

hankel2, besselj, bessely

References

[1]

http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.hankel2(nu, z)

Bases: BesselBase

Hankel function of the second kind.

This function is defined as

\[H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind, and \(Y_\nu(z)\) is the Bessel function of the second kind.

It is a solution to Bessel’s equation, and linearly independent from \(H_\nu^{(1)}\).

Examples

>>> from ... import hankel2
>>> from ...abc import z, n
>>> hankel2(n, z).diff(z)
hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2

See also

hankel1, besselj, bessely

References

[1]

http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.hn1(nu, z)

Bases: SphericalHankelBase

Spherical Hankel function of the first kind.

This function is defined as

\[h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z),\]

where \(j_\nu(z)\) and \(y_\nu(z)\) are the spherical Bessel function of the first and second kinds.

For integral orders \(n\), \(h_n^(1)\) is calculated using the formula:

\[h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)\]

Examples

>>> from ... import Symbol, hn1, hankel1, expand_func, yn, jn
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(hn1(nu, z)))
jn(nu, z) + I*yn(nu, z)
>>> print(expand_func(hn1(0, z)))
sin(z)/z - I*cos(z)/z
>>> print(expand_func(hn1(1, z)))
-I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2
>>> hn1(nu, z).rewrite(jn)
(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)
>>> hn1(nu, z).rewrite(yn)
(-1)**nu*yn(-nu - 1, z) + I*yn(nu, z)
>>> hn1(nu, z).rewrite(hankel1)
sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2

See also

hn2, jn, yn, hankel1, hankel2

References

[1]

http://dlmf.nist.gov/10.47

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.hn2(nu, z)

Bases: SphericalHankelBase

Spherical Hankel function of the second kind.

This function is defined as

\[h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z),\]

where \(j_\nu(z)\) and \(y_\nu(z)\) are the spherical Bessel function of the first and second kinds.

For integral orders \(n\), \(h_n^(2)\) is calculated using the formula:

\[h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)\]

Examples

>>> from ... import Symbol, hn2, hankel2, expand_func, jn, yn
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(hn2(nu, z)))
jn(nu, z) - I*yn(nu, z)
>>> print(expand_func(hn2(0, z)))
sin(z)/z + I*cos(z)/z
>>> print(expand_func(hn2(1, z)))
I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2
>>> hn2(nu, z).rewrite(hankel2)
sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2
>>> hn2(nu, z).rewrite(jn)
-(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)
>>> hn2(nu, z).rewrite(yn)
(-1)**nu*yn(-nu - 1, z) - I*yn(nu, z)

See also

hn1, jn, yn, hankel1, hankel2

References

[1]

http://dlmf.nist.gov/10.47

default_assumptions = {}

class modelparameters.sympy.functions.special.bessel.jn(nu, z)

Bases: SphericalBesselBase

Spherical Bessel function of the first kind.

This function is a solution to the spherical Bessel equation

\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\]

It can be defined as

\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\]

where \(J_\nu(z)\) is the Bessel function of the first kind.

The spherical Bessel functions of integral order are calculated using the formula:

\[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\]

where the coefficients \(f_n(z)\) are available as polys.orthopolys.spherical_bessel_fn().

Examples

>>> from ... import Symbol, jn, sin, cos, expand_func, besselj, bessely
>>> from ... import simplify
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(jn(0, z)))
sin(z)/z
>>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z
True
>>> expand_func(jn(3, z))
(-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)
>>> jn(nu, z).rewrite(besselj)
sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2
>>> jn(nu, z).rewrite(bessely)
(-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2
>>> jn(2, 5.2+0.3j).evalf(20)
0.099419756723640344491 - 0.054525080242173562897*I

See also

besselj, bessely, besselk, yn

References

[1]

http://dlmf.nist.gov/10.47

default_assumptions = {}

modelparameters.sympy.functions.special.bessel.jn_zeros(n, k, method='sympy', dps=15)

Zeros of the spherical Bessel function of the first kind.

This returns an array of zeros of jn up to the k-th zero.

• method = “sympy”: uses mpmath.besseljzero()

• method = “scipy”: uses the SciPy’s sph_jn and newton to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. [The function used with method=”sympy” is a recent addition to mpmath, before that a general solver was used.]

Examples

>>> from ... import jn_zeros
>>> jn_zeros(2, 4, dps=5)
[5.7635, 9.095, 12.323, 15.515]

See also

jn, yn, besselj, besselk, bessely

class modelparameters.sympy.functions.special.bessel.yn(nu, z)

Bases: SphericalBesselBase

Spherical Bessel function of the second kind.

This function is another solution to the spherical Bessel equation, and linearly independent from \(j_n\). It can be defined as

\[y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\]

where \(Y_\nu(z)\) is the Bessel function of the second kind.

For integral orders \(n\), \(y_n\) is calculated using the formula:

\[y_n(z) = (-1)^{n+1} j_{-n-1}(z)\]

Examples

>>> from ... import Symbol, yn, sin, cos, expand_func, besselj, bessely
>>> z = Symbol("z")
>>> nu = Symbol("nu", integer=True)
>>> print(expand_func(yn(0, z)))
-cos(z)/z
>>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z
True
>>> yn(nu, z).rewrite(besselj)
(-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2
>>> yn(nu, z).rewrite(bessely)
sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2
>>> yn(2, 5.2+0.3j).evalf(20)
0.18525034196069722536 + 0.014895573969924817587*I

See also

besselj, bessely, besselk, jn

References

[1]

http://dlmf.nist.gov/10.47

default_assumptions = {}

modelparameters.sympy.functions.special.beta_functions module

class modelparameters.sympy.functions.special.beta_functions.beta(x, y)

Bases: Function

The beta integral is called the Eulerian integral of the first kind by Legendre:

\[\mathrm{B}(x,y) := \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.\]

Beta function or Euler’s first integral is closely associated with gamma function. The Beta function often used in probability theory and mathematical statistics. It satisfies properties like:

\[\begin{split}\mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\end{split}\]

Therefore for integral values of a and b:

\[\mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}\]

Examples

>>> from ... import I, pi
>>> from ...abc import x,y

The Beta function obeys the mirror symmetry:

>>> from ... import beta
>>> from ... import conjugate
>>> conjugate(beta(x,y))
beta(conjugate(x), conjugate(y))

Differentiation with respect to both x and y is supported:

>>> from ... import beta
>>> from ... import diff
>>> diff(beta(x,y), x)
(polygamma(0, x) - polygamma(0, x + y))*beta(x, y)

>>> from ... import beta
>>> from ... import diff
>>> diff(beta(x,y), y)
(polygamma(0, y) - polygamma(0, x + y))*beta(x, y)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> from ... import beta
>>> beta(pi,pi).evalf(40)
0.02671848900111377452242355235388489324562

>>> beta(1+I,1+I).evalf(20)
-0.2112723729365330143 - 0.7655283165378005676*I

See also

sympy.functions.special.gamma_functions.gamma

Gamma function.

sympy.functions.special.gamma_functions.uppergamma

Upper incomplete gamma function.

sympy.functions.special.gamma_functions.lowergamma

Lower incomplete gamma function.

sympy.functions.special.gamma_functions.polygamma

Polygamma function.

sympy.functions.special.gamma_functions.loggamma

Log Gamma function.

sympy.functions.special.gamma_functions.digamma

Digamma function.

sympy.functions.special.gamma_functions.trigamma

Trigamma function.

References

[1]

http://en.wikipedia.org/wiki/Beta_function

[2]

http://mathworld.wolfram.com/BetaFunction.html

[3]

http://dlmf.nist.gov/5.12

default_assumptions = {}

classmethod eval(x, y)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex)

Returns the first derivative of the function.

nargs = {2}

unbranched = True

modelparameters.sympy.functions.special.bsplines module

modelparameters.sympy.functions.special.bsplines.bspline_basis(d, knots, n, x, close=True)

The n-th B-spline at x of degree d with knots.

B-Splines are piecewise polynomials of degree d [1]_. They are defined on a set of knots, which is a sequence of integers or floats.

The 0th degree splines have a value of one on a single interval:

>>> from ... import bspline_basis
>>> from ...abc import x
>>> d = 0
>>> knots = range(5)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, (x >= 0) & (x <= 1)), (0, True))

For a given (d, knots) there are len(knots)-d-1 B-splines defined, that are indexed by n (starting at 0).

Here is an example of a cubic B-spline:

>>> bspline_basis(3, range(5), 0, x)
Piecewise((x**3/6, (x >= 0) & (x < 1)),
          (-x**3/2 + 2*x**2 - 2*x + 2/3,
          (x >= 1) & (x < 2)),
          (x**3/2 - 4*x**2 + 10*x - 22/3,
          (x >= 2) & (x < 3)),
          (-x**3/6 + 2*x**2 - 8*x + 32/3,
          (x >= 3) & (x <= 4)),
          (0, True))

By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives:

>>> d = 1
>>> knots = [0,0,2,3,4]
>>> bspline_basis(d, knots, 0, x)
Piecewise((-x/2 + 1, (x >= 0) & (x <= 2)), (0, True))

It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-splines many times, it is best to lambdify them first:

>>> from ... import lambdify
>>> d = 3
>>> knots = range(10)
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)

See also

bsplines_basis_set

References

[1]

http://en.wikipedia.org/wiki/B-spline

modelparameters.sympy.functions.special.bsplines.bspline_basis_set(d, knots, x)

Return the len(knots)-d-1 B-splines at x of degree d with knots.

This function returns a list of Piecewise polynomials that are the len(knots)-d-1 B-splines of degree d for the given knots. This function calls bspline_basis(d, knots, n, x) for different values of n.

Examples

>>> from ... import bspline_basis_set
>>> from ...abc import x
>>> d = 2
>>> knots = range(5)
>>> splines = bspline_basis_set(d, knots, x)
>>> splines
[Piecewise((x**2/2, (x >= 0) & (x < 1)),
           (-x**2 + 3*x - 3/2, (x >= 1) & (x < 2)),
           (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
           (0, True)),
Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x < 2)),
          (-x**2 + 5*x - 11/2, (x >= 2) & (x < 3)),
          (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
          (0, True))]

See also

bsplines_basis

modelparameters.sympy.functions.special.delta_functions module

class modelparameters.sympy.functions.special.delta_functions.DiracDelta(arg, k=0)

Bases: Function

The DiracDelta function and its derivatives.

DiracDelta is not an ordinary function. It can be rigorously defined either as a distribution or as a measure.

DiracDelta only makes sense in definite integrals, and in particular, integrals of the form Integral(f(x)*DiracDelta(x - x0), (x, a, b)), where it equals f(x0) if a <= x0 <= b and 0 otherwise. Formally, DiracDelta acts in some ways like a function that is 0 everywhere except at 0, but in many ways it also does not. It can often be useful to treat DiracDelta in formal ways, building up and manipulating expressions with delta functions (which may eventually be integrated), but care must be taken to not treat it as a real function. SymPy’s oo is similar. It only truly makes sense formally in certain contexts (such as integration limits), but SymPy allows its use everywhere, and it tries to be consistent with operations on it (like 1/oo), but it is easy to get into trouble and get wrong results if oo is treated too much like a number. Similarly, if DiracDelta is treated too much like a function, it is easy to get wrong or nonsensical results.

DiracDelta function has the following properties:

1. diff(Heaviside(x),x) = DiracDelta(x)

2. integrate(DiracDelta(x-a)*f(x),(x,-oo,oo)) = f(a) and integrate(DiracDelta(x-a)*f(x),(x,a-e,a+e)) = f(a)

3. DiracDelta(x) = 0 for all x != 0

4. DiracDelta(g(x)) = Sum_i(DiracDelta(x-x_i)/abs(g'(x_i))) Where x_i-s are the roots of g

Derivatives of k-th order of DiracDelta have the following property:

5. DiracDelta(x,k) = 0, for all x != 0

Examples

>>> from ... import DiracDelta, diff, pi, Piecewise
>>> from ...abc import x, y

>>> DiracDelta(x)
DiracDelta(x)
>>> DiracDelta(1)
0
>>> DiracDelta(-1)
0
>>> DiracDelta(pi)
0
>>> DiracDelta(x - 4).subs(x, 4)
DiracDelta(0)
>>> diff(DiracDelta(x))
DiracDelta(x, 1)
>>> diff(DiracDelta(x - 1),x,2)
DiracDelta(x - 1, 2)
>>> diff(DiracDelta(x**2 - 1),x,2)
2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))
>>> DiracDelta(3*x).is_simple(x)
True
>>> DiracDelta(x**2).is_simple(x)
False
>>> DiracDelta((x**2 - 1)*y).expand(diracdelta=True, wrt=x)
DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))

See also

Heaviside, simplify, is_simple, sympy.functions.special.tensor_functions.KroneckerDelta

References

[1]

http://mathworld.wolfram.com/DeltaFunction.html

default_assumptions = {'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'real': True}

classmethod eval(arg, k=0)

Returns a simplified form or a value of DiracDelta depending on the argument passed by the DiracDelta object.

The eval() method is automatically called when the DiracDelta class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from ... import DiracDelta, S, Subs
>>> from ...abc import x

>>> DiracDelta(x)
DiracDelta(x)

>>> DiracDelta(x,1)
DiracDelta(x, 1)

>>> DiracDelta(1)
0

>>> DiracDelta(5,1)
0

>>> DiracDelta(0)
DiracDelta(0)

>>> DiracDelta(-1)
0

>>> DiracDelta(S.NaN)
nan

>>> DiracDelta(x).eval(1)
0

>>> DiracDelta(x - 100).subs(x, 5)
0

>>> DiracDelta(x - 100).subs(x, 100)
DiracDelta(0)

fdiff(argindex=1)

Returns the first derivative of a DiracDelta Function.

The difference between diff() and fdiff() is:- diff() is the user-level function and fdiff() is an object method. fdiff() is just a convenience method available in the Function class. It returns the derivative of the function without considering the chain rule. diff(function, x) calls Function._eval_derivative which in turn calls fdiff() internally to compute the derivative of the function.

Examples

>>> from ... import DiracDelta, diff
>>> from ...abc import x

>>> DiracDelta(x).fdiff()
DiracDelta(x, 1)

>>> DiracDelta(x, 1).fdiff()
DiracDelta(x, 2)

>>> DiracDelta(x**2 - 1).fdiff()
DiracDelta(x**2 - 1, 1)

>>> diff(DiracDelta(x, 1)).fdiff()
DiracDelta(x, 3)

is_commutative = True

is_complex = True

is_hermitian = True

is_imaginary = False

is_real = True

is_simple(self, x)

Tells whether the argument(args[0]) of DiracDelta is a linear expression in x.

x can be:

• a symbol

Examples

>>> from ... import DiracDelta, cos
>>> from ...abc import x, y

>>> DiracDelta(x*y).is_simple(x)
True
>>> DiracDelta(x*y).is_simple(y)
True

>>> DiracDelta(x**2 + x - 2).is_simple(x)
False

>>> DiracDelta(cos(x)).is_simple(x)
False

See also

simplify, Diracdelta

simplify(x)

See the simplify function in sympy.simplify

class modelparameters.sympy.functions.special.delta_functions.Heaviside(arg, H0=None)

Bases: Function

Heaviside Piecewise function

Heaviside function has the following properties [1]_:

1. diff(Heaviside(x),x) = DiracDelta(x)

   ( 0, if x < 0

2. Heaviside(x) = < ( undefined if x==0 [1]

   ( 1, if x > 0

3. Max(0,x).diff(x) = Heaviside(x)

[1]

Regarding to the value at 0, Mathematica defines H(0) = 1, but Maple uses H(0) = undefined. Different application areas may have specific conventions. For example, in control theory, it is common practice to assume H(0) == 0 to match the Laplace transform of a DiracDelta distribution.

To specify the value of Heaviside at x=0, a second argument can be given. Omit this 2nd argument or pass None to recover the default behavior.

>>> from ... import Heaviside, S
>>> from ...abc import x
>>> Heaviside(9)
1
>>> Heaviside(-9)
0
>>> Heaviside(0)
Heaviside(0)
>>> Heaviside(0, S.Half)
1/2
>>> (Heaviside(x) + 1).replace(Heaviside(x), Heaviside(x, 1))
Heaviside(x, 1) + 1

See also

DiracDelta

References

[2]

http://mathworld.wolfram.com/HeavisideStepFunction.html

[3]

http://dlmf.nist.gov/1.16#iv

default_assumptions = {'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'real': True}

classmethod eval(arg, H0=None)

Returns a simplified form or a value of Heaviside depending on the argument passed by the Heaviside object.

The eval() method is automatically called when the Heaviside class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from ... import Heaviside, S
>>> from ...abc import x

>>> Heaviside(x)
Heaviside(x)

>>> Heaviside(19)
1

>>> Heaviside(0)
Heaviside(0)

>>> Heaviside(0, 1)
1

>>> Heaviside(-5)
0

>>> Heaviside(S.NaN)
nan

>>> Heaviside(x).eval(100)
1

>>> Heaviside(x - 100).subs(x, 5)
0

>>> Heaviside(x - 100).subs(x, 105)
1

fdiff(argindex=1)

Returns the first derivative of a Heaviside Function.

Examples

>>> from ... import Heaviside, diff
>>> from ...abc import x

>>> Heaviside(x).fdiff()
DiracDelta(x)

>>> Heaviside(x**2 - 1).fdiff()
DiracDelta(x**2 - 1)

>>> diff(Heaviside(x)).fdiff()
DiracDelta(x, 1)

is_commutative = True

is_complex = True

is_hermitian = True

is_imaginary = False

is_real = True

modelparameters.sympy.functions.special.elliptic_integrals module

Elliptic integrals.

class modelparameters.sympy.functions.special.elliptic_integrals.elliptic_e(m, z=None)

Bases: Function

Called with two arguments z and m, evaluates the incomplete elliptic integral of the second kind, defined by

\[E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt\]

Called with a single argument m, evaluates the Legendre complete elliptic integral of the second kind

\[E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)\]

The function E(m) is a single-valued function on the complex plane with branch cut along the interval (1, infty).

Note that our notation defines the incomplete elliptic integral in terms of the parameter m instead of the elliptic modulus (eccentricity) k. In this case, the parameter m is defined as m=k^2.

Examples

>>> from ... import elliptic_e, I, pi, O
>>> from ...abc import z, m
>>> elliptic_e(z, m).series(z)
z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
>>> elliptic_e(m).series(n=4)
pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)
>>> elliptic_e(1 + I, 2 - I/2).n()
1.55203744279187 + 0.290764986058437*I
>>> elliptic_e(0)
pi/2
>>> elliptic_e(2.0 - I)
0.991052601328069 + 0.81879421395609*I

References

[1]

http://en.wikipedia.org/wiki/Elliptic_integrals

[2]

http://functions.wolfram.com/EllipticIntegrals/EllipticE2

[3]

http://functions.wolfram.com/EllipticIntegrals/EllipticE

default_assumptions = {}

classmethod eval(m, z=None)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.elliptic_integrals.elliptic_f(z, m)

Bases: Function

The Legendre incomplete elliptic integral of the first kind, defined by

\[F\left(z\middle| m\right) = \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}\]

This function reduces to a complete elliptic integral of the first kind, K(m), when z = pi/2.

Note that our notation defines the incomplete elliptic integral in terms of the parameter m instead of the elliptic modulus (eccentricity) k. In this case, the parameter m is defined as m=k^2.

Examples

>>> from ... import elliptic_f, I, O
>>> from ...abc import z, m
>>> elliptic_f(z, m).series(z)
z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
>>> elliptic_f(3.0 + I/2, 1.0 + I)
2.909449841483 + 1.74720545502474*I

References

[1]

http://en.wikipedia.org/wiki/Elliptic_integrals

[2]

http://functions.wolfram.com/EllipticIntegrals/EllipticF

See also

elliptic_k

default_assumptions = {}

classmethod eval(z, m)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.elliptic_integrals.elliptic_k(m)

Bases: Function

The complete elliptic integral of the first kind, defined by

\[K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)\]

where Fleft(zmiddle| mright) is the Legendre incomplete elliptic integral of the first kind.

The function K(m) is a single-valued function on the complex plane with branch cut along the interval (1, infty).

Note that our notation defines the incomplete elliptic integral in terms of the parameter m instead of the elliptic modulus (eccentricity) k. In this case, the parameter m is defined as m=k^2.

Examples

>>> from ... import elliptic_k, I, pi
>>> from ...abc import m
>>> elliptic_k(0)
pi/2
>>> elliptic_k(1.0 + I)
1.50923695405127 + 0.625146415202697*I
>>> elliptic_k(m).series(n=3)
pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)

References

[1]

http://en.wikipedia.org/wiki/Elliptic_integrals

[2]

http://functions.wolfram.com/EllipticIntegrals/EllipticK

See also

elliptic_f

default_assumptions = {}

classmethod eval(m)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.elliptic_integrals.elliptic_pi(n, m, z=None)

Bases: Function

Called with three arguments n, z and m, evaluates the Legendre incomplete elliptic integral of the third kind, defined by

\[\Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}\]

Called with two arguments n and m, evaluates the complete elliptic integral of the third kind:

\[\Pi\left(n\middle| m\right) = \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)\]

Note that our notation defines the incomplete elliptic integral in terms of the parameter m instead of the elliptic modulus (eccentricity) k. In this case, the parameter m is defined as m=k^2.

Examples

>>> from ... import elliptic_pi, I, pi, O, S
>>> from ...abc import z, n, m
>>> elliptic_pi(n, z, m).series(z, n=4)
z + z**3*(m/6 + n/3) + O(z**4)
>>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
2.50232379629182 - 0.760939574180767*I
>>> elliptic_pi(0, 0)
pi/2
>>> elliptic_pi(1.0 - I/3, 2.0 + I)
3.29136443417283 + 0.32555634906645*I

References

[1]

http://en.wikipedia.org/wiki/Elliptic_integrals

[2]

http://functions.wolfram.com/EllipticIntegrals/EllipticPi3

[3]

http://functions.wolfram.com/EllipticIntegrals/EllipticPi

default_assumptions = {}

classmethod eval(n, m, z=None)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.error_functions module

This module contains various functions that are special cases of incomplete gamma functions. It should probably be renamed.

class modelparameters.sympy.functions.special.error_functions.Chi(z)

Bases: TrigonometricIntegral

Cosh integral.

This function is defined for positive \(x\) by

\[\operatorname{Chi}(x) = \gamma + \log{x} + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,\]

where \(\gamma\) is the Euler-Mascheroni constant.

We have

\[\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right) - i\frac{\pi}{2},\]

which holds for all polar \(z\) and thus provides an analytic continuation to the Riemann surface of the logarithm. By lifting to the principal branch we obtain an analytic function on the cut complex plane.

Examples

>>> from ... import Chi
>>> from ...abc import z

The cosh integral is a primitive of cosh(z)/z:

>>> Chi(z).diff(z)
cosh(z)/z

It has a logarithmic branch point at the origin:

>>> from ... import exp_polar, I, pi
>>> Chi(z*exp_polar(2*I*pi))
Chi(z) + 2*I*pi

The cosh integral behaves somewhat like ordinary cosh under multiplication by i:

>>> from ... import polar_lift
>>> Chi(polar_lift(I)*z)
Ci(z) + I*pi/2
>>> Chi(polar_lift(-1)*z)
Chi(z) + I*pi

It can also be expressed in terms of exponential integrals:

>>> from ... import expint
>>> Chi(z).rewrite(expint)
-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

See also

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

References

[1]

http://en.wikipedia.org/wiki/Trigonometric_integral

default_assumptions = {}

class modelparameters.sympy.functions.special.error_functions.Ci(z)

Bases: TrigonometricIntegral

Cosine integral.

This function is defined for positive x by

\[\operatorname{Ci}(x) = \gamma + \log{x} + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,\]

where gamma is the Euler-Mascheroni constant.

We have

\[\operatorname{Ci}(z) = -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right) + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\]

which holds for all polar z and thus provides an analytic continuation to the Riemann surface of the logarithm.

The formula also holds as stated for z in mathbb{C} with Re(z) > 0. By lifting to the principal branch we obtain an analytic function on the cut complex plane.

Examples

>>> from ... import Ci
>>> from ...abc import z

The cosine integral is a primitive of cos(z)/z:

>>> Ci(z).diff(z)
cos(z)/z

It has a logarithmic branch point at the origin:

>>> from ... import exp_polar, I, pi
>>> Ci(z*exp_polar(2*I*pi))
Ci(z) + 2*I*pi

The cosine integral behaves somewhat like ordinary cos under multiplication by i:

>>> from ... import polar_lift
>>> Ci(polar_lift(I)*z)
Chi(z) + I*pi/2
>>> Ci(polar_lift(-1)*z)
Ci(z) + I*pi

It can also be expressed in terms of exponential integrals:

>>> from ... import expint
>>> Ci(z).rewrite(expint)
-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2

See also

Si

Sine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

References

[1]

http://en.wikipedia.org/wiki/Trigonometric_integral

default_assumptions = {}

modelparameters.sympy.functions.special.error_functions.E1(z)

Classical case of the generalized exponential integral.

This is equivalent to expint(1, z).

See also

Ei

Exponential integral.

expint

Generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

class modelparameters.sympy.functions.special.error_functions.Ei(z)

Bases: Function

The classical exponential integral.

For use in SymPy, this function is defined as

\[\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!} + \log(x) + \gamma,\]

where gamma is the Euler-Mascheroni constant.

If x is a polar number, this defines an analytic function on the Riemann surface of the logarithm. Otherwise this defines an analytic function in the cut plane mathbb{C} setminus (-infty, 0].

Background

The name exponential integral comes from the following statement:

\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\]

If the integral is interpreted as a Cauchy principal value, this statement holds for x > 0 and operatorname{Ei}(x) as defined above.

Note that we carefully avoided defining operatorname{Ei}(x) for negative real x. This is because above integral formula does not hold for any polar lift of such x, indeed all branches of operatorname{Ei}(x) above the negative reals are imaginary.

However, the following statement holds for all x in mathbb{R}^*:

\[\int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t = \frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) + \operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2},\]

where the integral is again understood to be a principal value if x > 0, and |x|e^{i arg(x)}, |x|e^{- i arg(x)} denote two conjugate polar lifts of x.

Examples

>>> from ... import Ei, polar_lift, exp_polar, I, pi
>>> from ...abc import x

The exponential integral in SymPy is strictly undefined for negative values of the argument. For convenience, exponential integrals with negative arguments are immediately converted into an expression that agrees with the classical integral definition:

>>> Ei(-1)
-I*pi + Ei(exp_polar(I*pi))

This yields a real value:

>>> Ei(-1).n(chop=True)
-0.219383934395520

On the other hand the analytic continuation is not real:

>>> Ei(polar_lift(-1)).n(chop=True)
-0.21938393439552 + 3.14159265358979*I

The exponential integral has a logarithmic branch point at the origin:

>>> Ei(x*exp_polar(2*I*pi))
Ei(x) + 2*I*pi

Differentiation is supported:

>>> Ei(x).diff(x)
exp(x)/x

The exponential integral is related to many other special functions. For example:

>>> from ... import uppergamma, expint, Shi
>>> Ei(x).rewrite(expint)
-expint(1, x*exp_polar(I*pi)) - I*pi
>>> Ei(x).rewrite(Shi)
Chi(x) + Shi(x)

See also

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

sympy.functions.special.gamma_functions.uppergamma

Upper incomplete gamma function.

References

[1]

http://dlmf.nist.gov/6.6

[2]

http://en.wikipedia.org/wiki/Exponential_integral

[3]

Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.FresnelIntegral(z)

Bases: Function

Base class for the Fresnel integrals.

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

unbranched = True

class modelparameters.sympy.functions.special.error_functions.Li(z)

Bases: Function

The offset logarithmic integral.

For the use in SymPy, this function is defined as

\[\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)\]

Examples

>>> from ... import I, oo, Li
>>> from ...abc import z

The following special value is known:

>>> Li(2)
0

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(Li(z), z)
1/log(z)

The shifted logarithmic integral can be written in terms of li(z):

>>> from ... import li
>>> Li(z).rewrite(li)
li(z) - li(2)

We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):

>>> Li(2).evalf(30)
0

>>> Li(4).evalf(30)
1.92242131492155809316615998938

See also

li

Logarithmic integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

References

[1]

http://en.wikipedia.org/wiki/Logarithmic_integral

[2]

http://mathworld.wolfram.com/LogarithmicIntegral.html

[3]

http://dlmf.nist.gov/6

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.Shi(z)

Bases: TrigonometricIntegral

Sinh integral.

This function is defined by

\[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from ... import Shi
>>> from ...abc import z

The Sinh integral is a primitive of sinh(z)/z:

>>> Shi(z).diff(z)
sinh(z)/z

It is unbranched:

>>> from ... import exp_polar, I, pi
>>> Shi(z*exp_polar(2*I*pi))
Shi(z)

The sinh integral behaves much like ordinary sinh under multiplication by i:

>>> Shi(I*z)
I*Si(z)
>>> Shi(-z)
-Shi(z)

It can also be expressed in terms of exponential integrals, but beware that the latter is branched:

>>> from ... import expint
>>> Shi(z).rewrite(expint)
expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2

See also

Si

Sine integral.

Ci

Cosine integral.

Chi

Hyperbolic cosine integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

References

[1]

http://en.wikipedia.org/wiki/Trigonometric_integral

default_assumptions = {}

class modelparameters.sympy.functions.special.error_functions.Si(z)

Bases: TrigonometricIntegral

Sine integral.

This function is defined by

\[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from ... import Si
>>> from ...abc import z

The sine integral is an antiderivative of sin(z)/z:

>>> Si(z).diff(z)
sin(z)/z

It is unbranched:

>>> from ... import exp_polar, I, pi
>>> Si(z*exp_polar(2*I*pi))
Si(z)

Sine integral behaves much like ordinary sine under multiplication by I:

>>> Si(I*z)
I*Shi(z)
>>> Si(-z)
-Si(z)

It can also be expressed in terms of exponential integrals, but beware that the latter is branched:

>>> from ... import expint
>>> Si(z).rewrite(expint)
-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +
     expint(1, z*exp_polar(I*pi/2))/2) + pi/2

It can be rewritten in the form of sinc function (By definition)

>>> from ... import sinc
>>> Si(z).rewrite(sinc)
Integral(sinc(t), (t, 0, z))

See also

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

sinc

unnormalized sinc function

E1

Special case of the generalised exponential integral.

li

Logarithmic integral.

Li

Offset logarithmic integral.

References

[1]

http://en.wikipedia.org/wiki/Trigonometric_integral

default_assumptions = {}

class modelparameters.sympy.functions.special.error_functions.TrigonometricIntegral(z)

Bases: Function

Base class for trigonometric integrals.

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.erf(arg)

Bases: Function

The Gauss error function. This function is defined as:

\[\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.\]

Examples

>>> from ... import I, oo, erf
>>> from ...abc import z

Several special values are known:

>>> erf(0)
0
>>> erf(oo)
1
>>> erf(-oo)
-1
>>> erf(I*oo)
oo*I
>>> erf(-I*oo)
-oo*I

In general one can pull out factors of -1 and I from the argument:

>>> erf(-z)
-erf(z)

The error function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(erf(z))
erf(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(erf(z), z)
2*exp(-z**2)/sqrt(pi)

We can numerically evaluate the error function to arbitrary precision on the whole complex plane:

>>> erf(4).evalf(30)
0.999999984582742099719981147840

>>> erf(-4*I).evalf(30)
-1296959.73071763923152794095062*I

See also

erfc

Complementary error function.

erfi

Imaginary error function.

erf2

Two-argument error function.

erfinv

Inverse error function.

erfcinv

Inverse Complementary error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://en.wikipedia.org/wiki/Error_function

[2]

http://dlmf.nist.gov/7

[3]

http://mathworld.wolfram.com/Erf.html

[4]

http://functions.wolfram.com/GammaBetaErf/Erf

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

inverse(argindex=1)

Returns the inverse of this function.

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

unbranched = True

class modelparameters.sympy.functions.special.error_functions.erf2(x, y)

Bases: Function

Two-argument error function. This function is defined as:

\[\mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t\]

Examples

>>> from ... import I, oo, erf2
>>> from ...abc import x, y

Several special values are known:

>>> erf2(0, 0)
0
>>> erf2(x, x)
0
>>> erf2(x, oo)
-erf(x) + 1
>>> erf2(x, -oo)
-erf(x) - 1
>>> erf2(oo, y)
erf(y) - 1
>>> erf2(-oo, y)
erf(y) + 1

In general one can pull out factors of -1:

>>> erf2(-x, -y)
-erf2(x, y)

The error function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(erf2(x, y))
erf2(conjugate(x), conjugate(y))

Differentiation with respect to x, y is supported:

>>> from ... import diff
>>> diff(erf2(x, y), x)
-2*exp(-x**2)/sqrt(pi)
>>> diff(erf2(x, y), y)
2*exp(-y**2)/sqrt(pi)

See also

erf

Gaussian error function.

erfc

Complementary error function.

erfi

Imaginary error function.

erfinv

Inverse error function.

erfcinv

Inverse Complementary error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://functions.wolfram.com/GammaBetaErf/Erf2/

default_assumptions = {}

classmethod eval(x, y)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.erf2inv(x, y)

Bases: Function

Two-argument Inverse error function. The erf2inv function is defined as:

\[\mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w\]

Examples

>>> from ... import I, oo, erf2inv, erfinv, erfcinv
>>> from ...abc import x, y

Several special values are known:

>>> erf2inv(0, 0)
0
>>> erf2inv(1, 0)
1
>>> erf2inv(0, 1)
oo
>>> erf2inv(0, y)
erfinv(y)
>>> erf2inv(oo, y)
erfcinv(-y)

Differentiation with respect to x and y is supported:

>>> from ... import diff
>>> diff(erf2inv(x, y), x)
exp(-x**2 + erf2inv(x, y)**2)
>>> diff(erf2inv(x, y), y)
sqrt(pi)*exp(erf2inv(x, y)**2)/2

See also

erf

Gaussian error function.

erfc

Complementary error function.

erfi

Imaginary error function.

erf2

Two-argument error function.

erfinv

Inverse error function.

erfcinv

Inverse complementary error function.

References

[1]

http://functions.wolfram.com/GammaBetaErf/InverseErf2/

default_assumptions = {}

classmethod eval(x, y)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.erfc(arg)

Bases: Function

Complementary Error Function. The function is defined as:

\[\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t\]

Examples

>>> from ... import I, oo, erfc
>>> from ...abc import z

Several special values are known:

>>> erfc(0)
1
>>> erfc(oo)
0
>>> erfc(-oo)
2
>>> erfc(I*oo)
-oo*I
>>> erfc(-I*oo)
oo*I

The error function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(erfc(z))
erfc(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(erfc(z), z)
-2*exp(-z**2)/sqrt(pi)

It also follows

>>> erfc(-z)
-erfc(z) + 2

We can numerically evaluate the complementary error function to arbitrary precision on the whole complex plane:

>>> erfc(4).evalf(30)
0.0000000154172579002800188521596734869

>>> erfc(4*I).evalf(30)
1.0 - 1296959.73071763923152794095062*I

See also

erf

Gaussian error function.

erfi

Imaginary error function.

erf2

Two-argument error function.

erfinv

Inverse error function.

erfcinv

Inverse Complementary error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://en.wikipedia.org/wiki/Error_function

[2]

http://dlmf.nist.gov/7

[3]

http://mathworld.wolfram.com/Erfc.html

[4]

http://functions.wolfram.com/GammaBetaErf/Erfc

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

inverse(argindex=1)

Returns the inverse of this function.

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

unbranched = True

class modelparameters.sympy.functions.special.error_functions.erfcinv(z)

Bases: Function

Inverse Complementary Error Function. The erfcinv function is defined as:

\[\mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x\]

Examples

>>> from ... import I, oo, erfcinv
>>> from ...abc import x

Several special values are known:

>>> erfcinv(1)
0
>>> erfcinv(0)
oo

Differentiation with respect to x is supported:

>>> from ... import diff
>>> diff(erfcinv(x), x)
-sqrt(pi)*exp(erfcinv(x)**2)/2

See also

erf

Gaussian error function.

erfc

Complementary error function.

erfi

Imaginary error function.

erf2

Two-argument error function.

erfinv

Inverse error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://en.wikipedia.org/wiki/Error_function#Inverse_functions

[2]

http://functions.wolfram.com/GammaBetaErf/InverseErfc/

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

inverse(argindex=1)

Returns the inverse of this function.

class modelparameters.sympy.functions.special.error_functions.erfi(z)

Bases: Function

Imaginary error function. The function erfi is defined as:

\[\mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t\]

Examples

>>> from ... import I, oo, erfi
>>> from ...abc import z

Several special values are known:

>>> erfi(0)
0
>>> erfi(oo)
oo
>>> erfi(-oo)
-oo
>>> erfi(I*oo)
I
>>> erfi(-I*oo)
-I

In general one can pull out factors of -1 and I from the argument:

>>> erfi(-z)
-erfi(z)

>>> from ... import conjugate
>>> conjugate(erfi(z))
erfi(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(erfi(z), z)
2*exp(z**2)/sqrt(pi)

We can numerically evaluate the imaginary error function to arbitrary precision on the whole complex plane:

>>> erfi(2).evalf(30)
18.5648024145755525987042919132

>>> erfi(-2*I).evalf(30)
-0.995322265018952734162069256367*I

See also

erf

Gaussian error function.

erfc

Complementary error function.

erf2

Two-argument error function.

erfinv

Inverse error function.

erfcinv

Inverse Complementary error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://en.wikipedia.org/wiki/Error_function

[2]

http://mathworld.wolfram.com/Erfi.html

[3]

http://functions.wolfram.com/GammaBetaErf/Erfi

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

unbranched = True

class modelparameters.sympy.functions.special.error_functions.erfinv(z)

Bases: Function

Inverse Error Function. The erfinv function is defined as:

\[\mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x\]

Examples

>>> from ... import I, oo, erfinv
>>> from ...abc import x

Several special values are known:

>>> erfinv(0)
0
>>> erfinv(1)
oo

Differentiation with respect to x is supported:

>>> from ... import diff
>>> diff(erfinv(x), x)
sqrt(pi)*exp(erfinv(x)**2)/2

We can numerically evaluate the inverse error function to arbitrary precision on [-1, 1]:

>>> erfinv(0.2).evalf(30)
0.179143454621291692285822705344

See also

erf

Gaussian error function.

erfc

Complementary error function.

erfi

Imaginary error function.

erf2

Two-argument error function.

erfcinv

Inverse Complementary error function.

erf2inv

Inverse two-argument error function.

References

[1]

http://en.wikipedia.org/wiki/Error_function#Inverse_functions

[2]

http://functions.wolfram.com/GammaBetaErf/InverseErf/

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

inverse(argindex=1)

Returns the inverse of this function.

class modelparameters.sympy.functions.special.error_functions.expint(nu, z)

Bases: Function

Generalized exponential integral.

This function is defined as

\[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\]

where Gamma(1 - nu, z) is the upper incomplete gamma function (uppergamma).

Hence for \(z\) with positive real part we have

\[\operatorname{E}_\nu(z) = \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t,\]

which explains the name.

The representation as an incomplete gamma function provides an analytic continuation for \(\operatorname{E}_\nu(z)\). If \(\nu\) is a non-positive integer the exponential integral is thus an unbranched function of \(z\), otherwise there is a branch point at the origin. Refer to the incomplete gamma function documentation for details of the branching behavior.

Examples

>>> from ... import expint, S
>>> from ...abc import nu, z

Differentiation is supported. Differentiation with respect to z explains further the name: for integral orders, the exponential integral is an iterated integral of the exponential function.

>>> expint(nu, z).diff(z)
-expint(nu - 1, z)

Differentiation with respect to nu has no classical expression:

>>> expint(nu, z).diff(nu)
-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, -nu + 1), ()), z)

At non-postive integer orders, the exponential integral reduces to the exponential function:

>>> expint(0, z)
exp(-z)/z
>>> expint(-1, z)
exp(-z)/z + exp(-z)/z**2

At half-integers it reduces to error functions:

>>> expint(S(1)/2, z)
sqrt(pi)*erfc(sqrt(z))/sqrt(z)

At positive integer orders it can be rewritten in terms of exponentials and expint(1, z). Use expand_func() to do this:

>>> from ... import expand_func
>>> expand_func(expint(5, z))
z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24

The generalised exponential integral is essentially equivalent to the incomplete gamma function:

>>> from ... import uppergamma
>>> expint(nu, z).rewrite(uppergamma)
z**(nu - 1)*uppergamma(-nu + 1, z)

As such it is branched at the origin:

>>> from ... import exp_polar, pi, I
>>> expint(4, z*exp_polar(2*pi*I))
I*pi*z**3/3 + expint(4, z)
>>> expint(nu, z*exp_polar(2*pi*I))
z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(-nu + 1) + expint(nu, z)

See also

Ei

Another related function called exponential integral.

E1

The classical case, returns expint(1, z).

li

Logarithmic integral.

Li

Offset logarithmic integral.

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

sympy.functions.special.gamma_functions.uppergamma

References

[1]

http://dlmf.nist.gov/8.19

[2]

http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/

[3]

http://en.wikipedia.org/wiki/Exponential_integral

default_assumptions = {}

classmethod eval(nu, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.error_functions.fresnelc(z)

Bases: FresnelIntegral

Fresnel integral C.

This function is defined by

\[\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from ... import I, oo, fresnelc
>>> from ...abc import z

Several special values are known:

>>> fresnelc(0)
0
>>> fresnelc(oo)
1/2
>>> fresnelc(-oo)
-1/2
>>> fresnelc(I*oo)
I/2
>>> fresnelc(-I*oo)
-I/2

In general one can pull out factors of -1 and i from the argument:

>>> fresnelc(-z)
-fresnelc(z)
>>> fresnelc(I*z)
I*fresnelc(z)

The Fresnel C integral obeys the mirror symmetry overline{C(z)} = C(bar{z}):

>>> from ... import conjugate
>>> conjugate(fresnelc(z))
fresnelc(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(fresnelc(z), z)
cos(pi*z**2/2)

Defining the Fresnel functions via an integral

>>> from ... import integrate, pi, cos, gamma, expand_func
>>> integrate(cos(pi*z**2/2), z)
fresnelc(z)*gamma(1/4)/(4*gamma(5/4))
>>> expand_func(integrate(cos(pi*z**2/2), z))
fresnelc(z)

We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:

>>> fresnelc(2).evalf(30)
0.488253406075340754500223503357

>>> fresnelc(-2*I).evalf(30)
-0.488253406075340754500223503357*I

See also

fresnels

Fresnel sine integral.

References

[1]

http://en.wikipedia.org/wiki/Fresnel_integral

[2]

http://dlmf.nist.gov/7

[3]

http://mathworld.wolfram.com/FresnelIntegrals.html

[4]

http://functions.wolfram.com/GammaBetaErf/FresnelC

[5]

The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley

default_assumptions = {}

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

class modelparameters.sympy.functions.special.error_functions.fresnels(z)

Bases: FresnelIntegral

Fresnel integral S.

This function is defined by

\[\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.\]

It is an entire function.

Examples

>>> from ... import I, oo, fresnels
>>> from ...abc import z

Several special values are known:

>>> fresnels(0)
0
>>> fresnels(oo)
1/2
>>> fresnels(-oo)
-1/2
>>> fresnels(I*oo)
-I/2
>>> fresnels(-I*oo)
I/2

In general one can pull out factors of -1 and i from the argument:

>>> fresnels(-z)
-fresnels(z)
>>> fresnels(I*z)
-I*fresnels(z)

The Fresnel S integral obeys the mirror symmetry overline{S(z)} = S(bar{z}):

>>> from ... import conjugate
>>> conjugate(fresnels(z))
fresnels(conjugate(z))

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(fresnels(z), z)
sin(pi*z**2/2)

Defining the Fresnel functions via an integral

>>> from ... import integrate, pi, sin, gamma, expand_func
>>> integrate(sin(pi*z**2/2), z)
3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))
>>> expand_func(integrate(sin(pi*z**2/2), z))
fresnels(z)

We can numerically evaluate the Fresnel integral to arbitrary precision on the whole complex plane:

>>> fresnels(2).evalf(30)
0.343415678363698242195300815958

>>> fresnels(-2*I).evalf(30)
0.343415678363698242195300815958*I

See also

fresnelc

Fresnel cosine integral.

References

[1]

http://en.wikipedia.org/wiki/Fresnel_integral

[2]

http://dlmf.nist.gov/7

[3]

http://mathworld.wolfram.com/FresnelIntegrals.html

[4]

http://functions.wolfram.com/GammaBetaErf/FresnelS

[5]

The converging factors for the fresnel integrals by John W. Wrench Jr. and Vicki Alley

default_assumptions = {}

static taylor_term(n, x, *previous_terms)

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

class modelparameters.sympy.functions.special.error_functions.li(z)

Bases: Function

The classical logarithmic integral.

For the use in SymPy, this function is defined as

\[\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.\]

Examples

>>> from ... import I, oo, li
>>> from ...abc import z

Several special values are known:

>>> li(0)
0
>>> li(1)
-oo
>>> li(oo)
oo

Differentiation with respect to z is supported:

>>> from ... import diff
>>> diff(li(z), z)
1/log(z)

Defining the li function via an integral:

The logarithmic integral can also be defined in terms of Ei:

>>> from ... import Ei
>>> li(z).rewrite(Ei)
Ei(log(z))
>>> diff(li(z).rewrite(Ei), z)
1/log(z)

We can numerically evaluate the logarithmic integral to arbitrary precision on the whole complex plane (except the singular points):

>>> li(2).evalf(30)
1.04516378011749278484458888919

>>> li(2*I).evalf(30)
1.0652795784357498247001125598 + 3.08346052231061726610939702133*I

We can even compute Soldner’s constant by the help of mpmath:

>>> from mpmath import findroot
>>> findroot(li, 2)
1.45136923488338

Further transformations include rewriting li in terms of the trigonometric integrals Si, Ci, Shi and Chi:

>>> from ... import Si, Ci, Shi, Chi
>>> li(z).rewrite(Si)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Ci)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Shi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
>>> li(z).rewrite(Chi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))

See also

Li

Offset logarithmic integral.

Ei

Exponential integral.

expint

Generalised exponential integral.

E1

Special case of the generalised exponential integral.

Si

Sine integral.

Ci

Cosine integral.

Shi

Hyperbolic sine integral.

Chi

Hyperbolic cosine integral.

References

[1]

http://en.wikipedia.org/wiki/Logarithmic_integral

[2]

http://mathworld.wolfram.com/LogarithmicIntegral.html

[3]

http://dlmf.nist.gov/6

[4]

http://mathworld.wolfram.com/SoldnersConstant.html

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.gamma_functions module

modelparameters.sympy.functions.special.gamma_functions.digamma(x)

The digamma function is the first derivative of the loggamma function i.e,

\[\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z) }\]

In this case, digamma(z) = polygamma(0, z).

See also

gamma

Gamma function.

lowergamma

Lower incomplete gamma function.

uppergamma

Upper incomplete gamma function.

polygamma

Polygamma function.

loggamma

Log Gamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Digamma_function

[2]

http://mathworld.wolfram.com/DigammaFunction.html

[3]

http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

class modelparameters.sympy.functions.special.gamma_functions.gamma(arg)

Bases: Function

The gamma function

\[\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{t} \mathrm{d}t.\]

The gamma function implements the function which passes through the values of the factorial function, i.e. Gamma(n) = (n - 1)! when n is an integer. More general, Gamma(z) is defined in the whole complex plane except at the negative integers where there are simple poles.

Examples

>>> from ... import S, I, pi, oo, gamma
>>> from ...abc import x

Several special values are known:

>>> gamma(1)
1
>>> gamma(4)
6
>>> gamma(S(3)/2)
sqrt(pi)/2

The Gamma function obeys the mirror symmetry:

>>> from ... import conjugate
>>> conjugate(gamma(x))
gamma(conjugate(x))

Differentiation with respect to x is supported:

>>> from ... import diff
>>> diff(gamma(x), x)
gamma(x)*polygamma(0, x)

Series expansion is also supported:

>>> from ... import series
>>> series(gamma(x), x, 0, 3)
1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> gamma(pi).evalf(40)
2.288037795340032417959588909060233922890
>>> gamma(1+I).evalf(20)
0.49801566811835604271 - 0.15494982830181068512*I

See also

lowergamma

Lower incomplete gamma function.

uppergamma

Upper incomplete gamma function.

polygamma

Polygamma function.

loggamma

Log Gamma function.

digamma

Digamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Gamma_function

[2]

http://dlmf.nist.gov/5

[3]

http://mathworld.wolfram.com/GammaFunction.html

[4]

http://functions.wolfram.com/GammaBetaErf/Gamma/

default_assumptions = {}

classmethod eval(arg)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

unbranched = True

class modelparameters.sympy.functions.special.gamma_functions.loggamma(z)

Bases: Function

The loggamma function implements the logarithm of the gamma function i.e, logGamma(x).

Examples

Several special values are known. For numerical integral arguments we have:

>>> from ... import loggamma
>>> loggamma(-2)
oo
>>> loggamma(0)
oo
>>> loggamma(1)
0
>>> loggamma(2)
0
>>> loggamma(3)
log(2)

and for symbolic values:

>>> from ... import Symbol
>>> n = Symbol("n", integer=True, positive=True)
>>> loggamma(n)
log(gamma(n))
>>> loggamma(-n)
oo

for half-integral values:

>>> from ... import S, pi
>>> loggamma(S(5)/2)
log(3*sqrt(pi)/4)
>>> loggamma(n/2)
log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))

and general rational arguments:

>>> from ... import expand_func
>>> L = loggamma(S(16)/3)
>>> expand_func(L).doit()
-5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)
>>> L = loggamma(S(19)/4)
>>> expand_func(L).doit()
-4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)
>>> L = loggamma(S(23)/7)
>>> expand_func(L).doit()
-3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)

The loggamma function has the following limits towards infinity:

>>> from ... import oo
>>> loggamma(oo)
oo
>>> loggamma(-oo)
zoo

The loggamma function obeys the mirror symmetry if x in mathbb{C} setminus {-infty, 0}:

>>> from ...abc import x
>>> from ... import conjugate
>>> conjugate(loggamma(x))
loggamma(conjugate(x))

Differentiation with respect to x is supported:

>>> from ... import diff
>>> diff(loggamma(x), x)
polygamma(0, x)

Series expansion is also supported:

>>> from ... import series
>>> series(loggamma(x), x, 0, 4)
-log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)

We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane:

>>> from ... import I
>>> loggamma(5).evalf(30)
3.17805383034794561964694160130
>>> loggamma(I).evalf(20)
-0.65092319930185633889 - 1.8724366472624298171*I

See also

gamma

Gamma function.

lowergamma

Lower incomplete gamma function.

uppergamma

Upper incomplete gamma function.

polygamma

Polygamma function.

digamma

Digamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Gamma_function

[2]

http://dlmf.nist.gov/5

[3]

http://mathworld.wolfram.com/LogGammaFunction.html

[4]

http://functions.wolfram.com/GammaBetaErf/LogGamma/

default_assumptions = {}

classmethod eval(z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.gamma_functions.lowergamma(a, x)

Bases: Function

The lower incomplete gamma function.

It can be defined as the meromorphic continuation of

\[\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).\]

This can be shown to be the same as

\[\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]

where \({}_1F_1\) is the (confluent) hypergeometric function.

Examples

>>> from ... import lowergamma, S
>>> from ...abc import s, x
>>> lowergamma(s, x)
lowergamma(s, x)
>>> lowergamma(3, x)
-x**2*exp(-x) - 2*x*exp(-x) + 2 - 2*exp(-x)
>>> lowergamma(-S(1)/2, x)
-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)

See also

gamma

Gamma function.

uppergamma

Upper incomplete gamma function.

polygamma

Polygamma function.

loggamma

Log Gamma function.

digamma

Digamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function

[2]

Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

[3]

http://dlmf.nist.gov/8

[4]

http://functions.wolfram.com/GammaBetaErf/Gamma2/

[5]

http://functions.wolfram.com/GammaBetaErf/Gamma3/

default_assumptions = {}

classmethod eval(a, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.gamma_functions.polygamma(n, z)

Bases: Function

The function polygamma(n, z) returns log(gamma(z)).diff(n + 1).

It is a meromorphic function on mathbb{C} and defined as the (n+1)-th derivative of the logarithm of the gamma function:

\[\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).\]

Examples

Several special values are known:

>>> from ... import S, polygamma
>>> polygamma(0, 1)
-EulerGamma
>>> polygamma(0, 1/S(2))
-2*log(2) - EulerGamma
>>> polygamma(0, 1/S(3))
-3*log(3)/2 - sqrt(3)*pi/6 - EulerGamma
>>> polygamma(0, 1/S(4))
-3*log(2) - pi/2 - EulerGamma
>>> polygamma(0, 2)
-EulerGamma + 1
>>> polygamma(0, 23)
-EulerGamma + 19093197/5173168

>>> from ... import oo, I
>>> polygamma(0, oo)
oo
>>> polygamma(0, -oo)
oo
>>> polygamma(0, I*oo)
oo
>>> polygamma(0, -I*oo)
oo

Differentiation with respect to x is supported:

>>> from ... import Symbol, diff
>>> x = Symbol("x")
>>> diff(polygamma(0, x), x)
polygamma(1, x)
>>> diff(polygamma(0, x), x, 2)
polygamma(2, x)
>>> diff(polygamma(0, x), x, 3)
polygamma(3, x)
>>> diff(polygamma(1, x), x)
polygamma(2, x)
>>> diff(polygamma(1, x), x, 2)
polygamma(3, x)
>>> diff(polygamma(2, x), x)
polygamma(3, x)
>>> diff(polygamma(2, x), x, 2)
polygamma(4, x)

>>> n = Symbol("n")
>>> diff(polygamma(n, x), x)
polygamma(n + 1, x)
>>> diff(polygamma(n, x), x, 2)
polygamma(n + 2, x)

We can rewrite polygamma functions in terms of harmonic numbers:

>>> from ... import harmonic
>>> polygamma(0, x).rewrite(harmonic)
harmonic(x - 1) - EulerGamma
>>> polygamma(2, x).rewrite(harmonic)
2*harmonic(x - 1, 3) - 2*zeta(3)
>>> ni = Symbol("n", integer=True)
>>> polygamma(ni, x).rewrite(harmonic)
(-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)

See also

gamma

Gamma function.

lowergamma

Lower incomplete gamma function.

uppergamma

Upper incomplete gamma function.

loggamma

Log Gamma function.

digamma

Digamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Polygamma_function

[2]

http://mathworld.wolfram.com/PolygammaFunction.html

[3]

http://functions.wolfram.com/GammaBetaErf/PolyGamma/

[4]

http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

default_assumptions = {}

classmethod eval(n, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.gamma_functions.trigamma(x)

The trigamma function is the second derivative of the loggamma function i.e,

\[\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).\]

In this case, trigamma(z) = polygamma(1, z).

See also

gamma

Gamma function.

lowergamma

Lower incomplete gamma function.

uppergamma

Upper incomplete gamma function.

polygamma

Polygamma function.

loggamma

Log Gamma function.

digamma

Digamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Trigamma_function

[2]

http://mathworld.wolfram.com/TrigammaFunction.html

[3]

http://functions.wolfram.com/GammaBetaErf/PolyGamma2/

class modelparameters.sympy.functions.special.gamma_functions.uppergamma(a, z)

Bases: Function

The upper incomplete gamma function.

It can be defined as the meromorphic continuation of

\[\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).\]

where gamma(s, x) is the lower incomplete gamma function, lowergamma. This can be shown to be the same as

\[\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]

where \({}_1F_1\) is the (confluent) hypergeometric function.

The upper incomplete gamma function is also essentially equivalent to the generalized exponential integral:

\[\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).\]

Examples

>>> from ... import uppergamma, S
>>> from ...abc import s, x
>>> uppergamma(s, x)
uppergamma(s, x)
>>> uppergamma(3, x)
x**2*exp(-x) + 2*x*exp(-x) + 2*exp(-x)
>>> uppergamma(-S(1)/2, x)
-2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)
>>> uppergamma(-2, x)
expint(3, x)/x**2

See also

gamma

Gamma function.

lowergamma

Lower incomplete gamma function.

polygamma

Polygamma function.

loggamma

Log Gamma function.

digamma

Digamma function.

trigamma

Trigamma function.

sympy.functions.special.beta_functions.beta

Euler Beta function.

References

[1]

http://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function

[2]

Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

[3]

http://dlmf.nist.gov/8

[4]

http://functions.wolfram.com/GammaBetaErf/Gamma2/

[5]

http://functions.wolfram.com/GammaBetaErf/Gamma3/

[6]

http://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions

default_assumptions = {}

classmethod eval(a, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.hyper module

Hypergeometric and Meijer G-functions

class modelparameters.sympy.functions.special.hyper.HyperRep(*args)

Bases: Function

A base class for “hyper representation functions”.

This is used exclusively in hyperexpand(), but fits more logically here.

pFq is branched at 1 if p == q+1. For use with slater-expansion, we want define an “analytic continuation” to all polar numbers, which is continuous on circles and on the ray t*exp_polar(I*pi). Moreover, we want a “nice” expression for the various cases.

This base class contains the core logic, concrete derived classes only supply the actual functions.

default_assumptions = {}

classmethod eval(*args)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.hyper.HyperRep_asin1(*args)

Bases: HyperRep

Represent hyper([1/2, 1/2], [3/2], z) == asin(sqrt(z))/sqrt(z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_asin2(*args)

Bases: HyperRep

Represent hyper([1, 1], [3/2], z) == asin(sqrt(z))/sqrt(z)/sqrt(1-z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_atanh(*args)

Bases: HyperRep

Represent hyper([1/2, 1], [3/2], z) == atanh(sqrt(z))/sqrt(z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_cosasin(*args)

Bases: HyperRep

Represent hyper([a, -a], [1/2], z) == cos(2*a*asin(sqrt(z))).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_log1(*args)

Bases: HyperRep

Represent -z*hyper([1, 1], [2], z) == log(1 - z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_log2(*args)

Bases: HyperRep

Represent log(1/2 + sqrt(1 - z)/2) == -z/4*hyper([3/2, 1, 1], [2, 2], z)

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_power1(*args)

Bases: HyperRep

Return a representative for hyper([-a], [], z) == (1 - z)**a.

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_power2(*args)

Bases: HyperRep

Return a representative for hyper([a, a - 1/2], [2*a], z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_sinasin(*args)

Bases: HyperRep

Represent 2*a*z*hyper([1 - a, 1 + a], [3/2], z) == sqrt(z)/sqrt(1-z)*sin(2*a*asin(sqrt(z)))

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_sqrts1(*args)

Bases: HyperRep

Return a representative for hyper([-a, 1/2 - a], [1/2], z).

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.HyperRep_sqrts2(*args)

Bases: HyperRep

Return a representative for sqrt(z)/2*[(1-sqrt(z))**2a - (1 + sqrt(z))**2a] == -2*z/(2*a+1) d/dz hyper([-a - 1/2, -a], [1/2], z)

default_assumptions = {}

class modelparameters.sympy.functions.special.hyper.TupleArg(*args, **kwargs)

Bases: Tuple

default_assumptions = {}

limit(x, xlim, dir='+')

Compute limit x->xlim.

class modelparameters.sympy.functions.special.hyper.TupleParametersBase(*args)

Bases: Function

Base class that takes care of differentiation, when some of the arguments are actually tuples.

default_assumptions = {'commutative': True}

is_commutative = True

class modelparameters.sympy.functions.special.hyper.hyper(ap, bq, z)

Bases: TupleParametersBase

The (generalized) hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain.

The hypergeometric function depends on two vectors of parameters, called the numerator parameters \(a_p\), and the denominator parameters \(b_q\). It also has an argument \(z\). The series definition is

\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix} \middle| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!},\end{split}\]

where \((a)_n = (a)(a+1)\cdots(a+n-1)\) denotes the rising factorial.

If one of the \(b_q\) is a non-positive integer then the series is undefined unless one of the a_p is a larger (i.e. smaller in magnitude) non-positive integer. If none of the \(b_q\) is a non-positive integer and one of the \(a_p\) is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the \(a_p\) or \(b_q\) is a non-positive integer. For more details, see the references.

The series converges for all \(z\) if \(p \le q\), and thus defines an entire single-valued function in this case. If \(p = q+1\) the series converges for \(|z| < 1\), and can be continued analytically into a half-plane. If \(p > q+1\) the series is divergent for all \(z\).

Note: The hypergeometric function constructor currently does not check if the parameters actually yield a well-defined function.

Examples

The parameters \(a_p\) and \(b_q\) can be passed as arbitrary iterables, for example:

>>> from ...functions import hyper
>>> from ...abc import x, n, a
>>> hyper((1, 2, 3), [3, 4], x)
hyper((1, 2, 3), (3, 4), x)

There is also pretty printing (it looks better using unicode):

>>> from ... import pprint
>>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False)
  _
 |_  /1, 2, 3 |  \
 |   |        | x|
3  2 \  3, 4  |  /

The parameters must always be iterables, even if they are vectors of length one or zero:

>>> hyper((1, ), [], x)
hyper((1,), (), x)

But of course they may be variables (but if they depend on x then you should not expect much implemented functionality):

>>> hyper((n, a), (n**2,), x)
hyper((n, a), (n**2,), x)

The hypergeometric function generalizes many named special functions. The function hyperexpand() tries to express a hypergeometric function using named special functions. For example:

>>> from ... import hyperexpand
>>> hyperexpand(hyper([], [], x))
exp(x)

You can also use expand_func:

>>> from ... import expand_func
>>> expand_func(x*hyper([1, 1], [2], -x))
log(x + 1)

More examples:

>>> from ... import S
>>> hyperexpand(hyper([], [S(1)/2], -x**2/4))
cos(x)
>>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2))
asin(x)

We can also sometimes hyperexpand parametric functions:

>>> from ...abc import a
>>> hyperexpand(hyper([-a], [], x))
(-x + 1)**a

See also

sympy.simplify.hyperexpand, sympy.functions.special.gamma_functions.gamma, meijerg

References

[1]

Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1

[2]

http://en.wikipedia.org/wiki/Generalized_hypergeometric_function

property ap

Numerator parameters of the hypergeometric function.

property argument

Argument of the hypergeometric function.

property bq

Denominator parameters of the hypergeometric function.

property convergence_statement

Return a condition on z under which the series converges.

default_assumptions = {'commutative': True}

property eta

A quantity related to the convergence of the series.

classmethod eval(ap, bq, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=3)

Returns the first derivative of the function.

is_commutative = True

property radius_of_convergence

Compute the radius of convergence of the defining series.

Note that even if this is not oo, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else.

>>> from ...functions import hyper
>>> from ...abc import z
>>> hyper((1, 2), [3], z).radius_of_convergence
1
>>> hyper((1, 2, 3), [4], z).radius_of_convergence
0
>>> hyper((1, 2), (3, 4), z).radius_of_convergence
oo

class modelparameters.sympy.functions.special.hyper.meijerg(*args)

Bases: TupleParametersBase

The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalizes the hypergeometric functions.

The Meijer G-function depends on four sets of parameters. There are “numerator parameters” \(a_1, \ldots, a_n\) and \(a_{n+1}, \ldots, a_p\), and there are “denominator parameters” \(b_1, \ldots, b_m\) and \(b_{m+1}, \ldots, b_q\). Confusingly, it is traditionally denoted as follows (note the position of m, n, p, q, and how they relate to the lengths of the four parameter vectors):

\[\begin{split}G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\ b_1, \cdots, b_m & b_{m+1}, \cdots, b_q \end{matrix} \middle| z \right).\end{split}\]

However, in sympy the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol.

The G function is defined as the following integral:

\[\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]

where \(\Gamma(z)\) is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them the definitions agree. The contours all separate the poles of \(\Gamma(1-a_j+s)\) from the poles of \(\Gamma(b_k-s)\), so in particular the G function is undefined if \(a_j - b_k \in \mathbb{Z}_{>0}\) for some \(j \le n\) and \(k \le m\).

The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references.

Note: Currently the Meijer G-function constructor does not check any convergence conditions.

Examples

You can pass the parameters either as four separate vectors:

>>> from ...functions import meijerg
>>> from ...abc import x, a
>>> from ...core.containers import Tuple
>>> from ... import pprint
>>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False)
 __1, 2 /1, 2  a, 4 |  \
/__     |           | x|
\_|4, 1 \ 5         |  /

or as two nested vectors:

>>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False)
 __1, 2 /1, 2  3, 4 |  \
/__     |           | x|
\_|4, 1 \ 5         |  /

As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected.

All the subvectors of parameters are available:

>>> from ... import pprint
>>> g = meijerg([1], [2], [3], [4], x)
>>> pprint(g, use_unicode=False)
 __1, 1 /1  2 |  \
/__     |     | x|
\_|2, 2 \3  4 |  /
>>> g.an
(1,)
>>> g.ap
(1, 2)
>>> g.aother
(2,)
>>> g.bm
(3,)
>>> g.bq
(3, 4)
>>> g.bother
(4,)

The Meijer G-function generalizes the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater’s theorem. For example:

>>> from ... import hyperexpand
>>> from ...abc import a, b, c
>>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True)
x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),
                             (-b + c + 1,), -x)/gamma(-b + c + 1)

Thus the Meijer G-function also subsumes many named functions as special cases. You can use expand_func or hyperexpand to (try to) rewrite a Meijer G-function in terms of named special functions. For example:

>>> from ... import expand_func, S
>>> expand_func(meijerg([[],[]], [[0],[]], -x))
exp(x)
>>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2))
sin(x)/sqrt(pi)

See also

hyper, sympy.simplify.hyperexpand

References

[1]

Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1

[2]

http://en.wikipedia.org/wiki/Meijer_G-function

property an

First set of numerator parameters.

property aother

Second set of numerator parameters.

property ap

Combined numerator parameters.

property argument

Argument of the Meijer G-function.

property bm

First set of denominator parameters.

property bother

Second set of denominator parameters.

property bq

Combined denominator parameters.

default_assumptions = {'commutative': True}

property delta

A quantity related to the convergence region of the integral, c.f. references.

fdiff(argindex=3)

Returns the first derivative of the function.

get_period()

Return a number P such that G(x*exp(I*P)) == G(x).

>>> from .hyper import meijerg
>>> from ...abc import z
>>> from ... import pi, S

>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
12*pi

integrand(s)

Get the defining integrand D(s).

is_commutative = True

property nu

A quantity related to the convergence region of the integral, c.f. references.

modelparameters.sympy.functions.special.mathieu_functions module

This module contains the Mathieu functions.

class modelparameters.sympy.functions.special.mathieu_functions.MathieuBase(*args)

Bases: Function

Abstract base class for Mathieu functions.

This class is meant to reduce code duplication.

default_assumptions = {}

unbranched = True

class modelparameters.sympy.functions.special.mathieu_functions.mathieuc(a, q, z)

Bases: MathieuBase

The Mathieu Cosine function C(a,q,z). This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Sine function.

Examples

>>> from ... import diff, mathieuc
>>> from ...abc import a, q, z

>>> mathieuc(a, q, z)
mathieuc(a, q, z)

>>> mathieuc(a, 0, z)
cos(sqrt(a)*z)

>>> diff(mathieuc(a, q, z), z)
mathieucprime(a, q, z)

See also

mathieus

Mathieu sine function

mathieusprime

Derivative of Mathieu sine function

mathieucprime

Derivative of Mathieu cosine function

References

[1]

http://en.wikipedia.org/wiki/Mathieu_function

[2]

http://dlmf.nist.gov/28

[3]

http://mathworld.wolfram.com/MathieuBase.html

[4]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/

default_assumptions = {}

classmethod eval(a, q, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.mathieu_functions.mathieucprime(a, q, z)

Bases: MathieuBase

The derivative C^{prime}(a,q,z) of the Mathieu Cosine function. This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Sine function.

Examples

>>> from ... import diff, mathieucprime
>>> from ...abc import a, q, z

>>> mathieucprime(a, q, z)
mathieucprime(a, q, z)

>>> mathieucprime(a, 0, z)
-sqrt(a)*sin(sqrt(a)*z)

>>> diff(mathieucprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieuc(a, q, z)

See also

mathieus

Mathieu sine function

mathieuc

Mathieu cosine function

mathieusprime

Derivative of Mathieu sine function

References

[1]

http://en.wikipedia.org/wiki/Mathieu_function

[2]

http://dlmf.nist.gov/28

[3]

http://mathworld.wolfram.com/MathieuBase.html

[4]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/

default_assumptions = {}

classmethod eval(a, q, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.mathieu_functions.mathieus(a, q, z)

Bases: MathieuBase

The Mathieu Sine function S(a,q,z). This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Cosine function.

Examples

>>> from ... import diff, mathieus
>>> from ...abc import a, q, z

>>> mathieus(a, q, z)
mathieus(a, q, z)

>>> mathieus(a, 0, z)
sin(sqrt(a)*z)

>>> diff(mathieus(a, q, z), z)
mathieusprime(a, q, z)

See also

mathieuc

Mathieu cosine function.

mathieusprime

Derivative of Mathieu sine function.

mathieucprime

Derivative of Mathieu cosine function.

References

[1]

http://en.wikipedia.org/wiki/Mathieu_function

[2]

http://dlmf.nist.gov/28

[3]

http://mathworld.wolfram.com/MathieuBase.html

[4]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/

default_assumptions = {}

classmethod eval(a, q, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.mathieu_functions.mathieusprime(a, q, z)

Bases: MathieuBase

The derivative S^{prime}(a,q,z) of the Mathieu Sine function. This function is one solution of the Mathieu differential equation:

\[y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0\]

The other solution is the Mathieu Cosine function.

Examples

>>> from ... import diff, mathieusprime
>>> from ...abc import a, q, z

>>> mathieusprime(a, q, z)
mathieusprime(a, q, z)

>>> mathieusprime(a, 0, z)
sqrt(a)*cos(sqrt(a)*z)

>>> diff(mathieusprime(a, q, z), z)
(-a + 2*q*cos(2*z))*mathieus(a, q, z)

See also

mathieus

Mathieu sine function

mathieuc

Mathieu cosine function

mathieucprime

Derivative of Mathieu cosine function

References

[1]

http://en.wikipedia.org/wiki/Mathieu_function

[2]

http://dlmf.nist.gov/28

[3]

http://mathworld.wolfram.com/MathieuBase.html

[4]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/

default_assumptions = {}

classmethod eval(a, q, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.polynomials module

This module mainly implements special orthogonal polynomials.

See also functions.combinatorial.numbers which contains some combinatorial polynomials.

class modelparameters.sympy.functions.special.polynomials.OrthogonalPolynomial(*args)

Bases: Function

Base class for orthogonal polynomials.

default_assumptions = {}

class modelparameters.sympy.functions.special.polynomials.assoc_laguerre(n, alpha, x)

Bases: OrthogonalPolynomial

Returns the nth generalized Laguerre polynomial in x, \(L_n(x)\).

Parameters:

• n (int) – Degree of Laguerre polynomial. Must be n >= 0.

• alpha (Expr) – Arbitrary expression. For alpha=0 regular Laguerre polynomials will be generated.

Examples

>>> from ... import laguerre, assoc_laguerre, diff
>>> from ...abc import x, n, a
>>> assoc_laguerre(0, a, x)
1
>>> assoc_laguerre(1, a, x)
a - x + 1
>>> assoc_laguerre(2, a, x)
a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
>>> assoc_laguerre(3, a, x)
a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
    x*(-a**2/2 - 5*a/2 - 3) + 1

>>> assoc_laguerre(n, a, 0)
binomial(a + n, a)

>>> assoc_laguerre(n, a, x)
assoc_laguerre(n, a, x)

>>> assoc_laguerre(n, 0, x)
laguerre(n, x)

>>> diff(assoc_laguerre(n, a, x), x)
-assoc_laguerre(n - 1, a + 1, x)

>>> diff(assoc_laguerre(n, a, x), a)
Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials

[2]

http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html

[3]

http://functions.wolfram.com/Polynomials/LaguerreL/

[4]

http://functions.wolfram.com/Polynomials/LaguerreL3/

default_assumptions = {}

classmethod eval(n, alpha, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=3)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.assoc_legendre(n, m, x)

Bases: Function

assoc_legendre(n,m, x) gives \(P_n^m(x)\), where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, \(P_n(x)\) in the following manner:

\[P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}\]

Associated Legendre polynomial are orthogonal on [-1, 1] with:

• weight = 1 for the same m, and different n.

• weight = 1/(1-x**2) for the same n, and different m.

Examples

>>> from ... import assoc_legendre
>>> from ...abc import x, m, n
>>> assoc_legendre(0,0, x)
1
>>> assoc_legendre(1,0, x)
x
>>> assoc_legendre(1,1, x)
-sqrt(-x**2 + 1)
>>> assoc_legendre(n,m,x)
assoc_legendre(n, m, x)

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Associated_Legendre_polynomials

[2]

http://mathworld.wolfram.com/LegendrePolynomial.html

[3]

http://functions.wolfram.com/Polynomials/LegendreP/

[4]

http://functions.wolfram.com/Polynomials/LegendreP2/

default_assumptions = {}

classmethod eval(n, m, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=3)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.chebyshevt(n, x)

Bases: OrthogonalPolynomial

Chebyshev polynomial of the first kind, \(T_n(x)\)

chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first kind) in x, \(T_n(x)\).

The Chebyshev polynomials of the first kind are orthogonal on \([-1, 1]\) with respect to the weight \(\frac{1}{\sqrt{1-x^2}}\).

Examples

>>> from ... import chebyshevt, chebyshevu, diff
>>> from ...abc import n,x
>>> chebyshevt(0, x)
1
>>> chebyshevt(1, x)
x
>>> chebyshevt(2, x)
2*x**2 - 1

>>> chebyshevt(n, x)
chebyshevt(n, x)
>>> chebyshevt(n, -x)
(-1)**n*chebyshevt(n, x)
>>> chebyshevt(-n, x)
chebyshevt(n, x)

>>> chebyshevt(n, 0)
cos(pi*n/2)
>>> chebyshevt(n, -1)
(-1)**n

>>> diff(chebyshevt(n, x), x)
n*chebyshevu(n - 1, x)

See also

jacobi, gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Chebyshev_polynomial

[2]

http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html

[3]

http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

[4]

http://functions.wolfram.com/Polynomials/ChebyshevT/

[5]

http://functions.wolfram.com/Polynomials/ChebyshevU/

default_assumptions = {}

classmethod eval(n, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.chebyshevt_root(n, k)

Bases: Function

chebyshev_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0.

Examples

>>> from ... import chebyshevt, chebyshevt_root
>>> chebyshevt_root(3, 2)
-sqrt(3)/2
>>> chebyshevt(3, chebyshevt_root(3, 2))
0

See also

jacobi, gegenbauer, chebyshevt, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

default_assumptions = {}

classmethod eval(n, k)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.polynomials.chebyshevu(n, x)

Bases: OrthogonalPolynomial

Chebyshev polynomial of the second kind, \(U_n(x)\)

chebyshevu(n, x) gives the nth Chebyshev polynomial of the second kind in x, \(U_n(x)\).

The Chebyshev polynomials of the second kind are orthogonal on \([-1, 1]\) with respect to the weight \(\sqrt{1-x^2}\).

Examples

>>> from ... import chebyshevt, chebyshevu, diff
>>> from ...abc import n,x
>>> chebyshevu(0, x)
1
>>> chebyshevu(1, x)
2*x
>>> chebyshevu(2, x)
4*x**2 - 1

>>> chebyshevu(n, x)
chebyshevu(n, x)
>>> chebyshevu(n, -x)
(-1)**n*chebyshevu(n, x)
>>> chebyshevu(-n, x)
-chebyshevu(n - 2, x)

>>> chebyshevu(n, 0)
cos(pi*n/2)
>>> chebyshevu(n, 1)
n + 1

>>> diff(chebyshevu(n, x), x)
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Chebyshev_polynomial

[2]

http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html

[3]

http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

[4]

http://functions.wolfram.com/Polynomials/ChebyshevT/

[5]

http://functions.wolfram.com/Polynomials/ChebyshevU/

default_assumptions = {}

classmethod eval(n, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.chebyshevu_root(n, k)

Bases: Function

chebyshevu_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, chebyshevu(n, chebyshevu_root(n, k)) == 0.

Examples

>>> from ... import chebyshevu, chebyshevu_root
>>> chebyshevu_root(3, 2)
-sqrt(2)/2
>>> chebyshevu(3, chebyshevu_root(3, 2))
0

See also

chebyshevt, chebyshevt_root, chebyshevu, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

default_assumptions = {}

classmethod eval(n, k)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.polynomials.gegenbauer(n, a, x)

Bases: OrthogonalPolynomial

Gegenbauer polynomial \(C_n^{\left(\alpha\right)}(x)\)

gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial in x, \(C_n^{\left(\alpha\right)}(x)\).

The Gegenbauer polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\).

Examples

>>> from ... import gegenbauer, conjugate, diff
>>> from ...abc import n,a,x
>>> gegenbauer(0, a, x)
1
>>> gegenbauer(1, a, x)
2*a*x
>>> gegenbauer(2, a, x)
-a + x**2*(2*a**2 + 2*a)
>>> gegenbauer(3, a, x)
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)

>>> gegenbauer(n, a, x)
gegenbauer(n, a, x)
>>> gegenbauer(n, a, -x)
(-1)**n*gegenbauer(n, a, x)

>>> gegenbauer(n, a, 0)
2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(-n/2 + 1/2)*gamma(n + 1))
>>> gegenbauer(n, a, 1)
gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))

>>> conjugate(gegenbauer(n, a, x))
gegenbauer(n, conjugate(a), conjugate(x))

>>> diff(gegenbauer(n, a, x), x)
2*a*gegenbauer(n - 1, a + 1, x)

See also

jacobi, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Gegenbauer_polynomials

[2]

http://mathworld.wolfram.com/GegenbauerPolynomial.html

[3]

http://functions.wolfram.com/Polynomials/GegenbauerC3/

default_assumptions = {}

classmethod eval(n, a, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=3)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.hermite(n, x)

Bases: OrthogonalPolynomial

hermite(n, x) gives the nth Hermite polynomial in x, \(H_n(x)\)

The Hermite polynomials are orthogonal on \((-\infty, \infty)\) with respect to the weight \(\exp\left(-x^2\right)\).

Examples

>>> from ... import hermite, diff
>>> from ...abc import x, n
>>> hermite(0, x)
1
>>> hermite(1, x)
2*x
>>> hermite(2, x)
4*x**2 - 2
>>> hermite(n, x)
hermite(n, x)
>>> diff(hermite(n,x), x)
2*n*hermite(n - 1, x)
>>> hermite(n, -x)
(-1)**n*hermite(n, x)

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Hermite_polynomial

[2]

http://mathworld.wolfram.com/HermitePolynomial.html

[3]

http://functions.wolfram.com/Polynomials/HermiteH/

default_assumptions = {}

classmethod eval(n, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.jacobi(n, a, b, x)

Bases: OrthogonalPolynomial

Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)

jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).

The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).

Examples

>>> from ... import jacobi, S, conjugate, diff
>>> from ...abc import n,a,b,x

>>> jacobi(0, a, b, x)
1
>>> jacobi(1, a, b, x)
a/2 - b/2 + x*(a/2 + b/2 + 1)
>>> jacobi(2, a, b, x)   
(a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +
b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2)

>>> jacobi(n, a, b, x)
jacobi(n, a, b, x)

>>> jacobi(n, a, a, x)
RisingFactorial(a + 1, n)*gegenbauer(n,
    a + 1/2, x)/RisingFactorial(2*a + 1, n)

>>> jacobi(n, 0, 0, x)
legendre(n, x)

>>> jacobi(n, S(1)/2, S(1)/2, x)
RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)

>>> jacobi(n, -S(1)/2, -S(1)/2, x)
RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)

>>> jacobi(n, a, b, -x)
(-1)**n*jacobi(n, b, a, x)

>>> jacobi(n, a, b, 0)
2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
>>> jacobi(n, a, b, 1)
RisingFactorial(a + 1, n)/factorial(n)

>>> conjugate(jacobi(n, a, b, x))
jacobi(n, conjugate(a), conjugate(b), conjugate(x))

>>> diff(jacobi(n,a,b,x), x)
(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)

See also

gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Jacobi_polynomials

[2]

http://mathworld.wolfram.com/JacobiPolynomial.html

[3]

http://functions.wolfram.com/Polynomials/JacobiP/

default_assumptions = {}

classmethod eval(n, a, b, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=4)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.polynomials.jacobi_normalized(n, a, b, x)

Jacobi polynomial \(P_n^{\left(\alpha, \beta\right)}(x)\)

jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial in x, \(P_n^{\left(\alpha, \beta\right)}(x)\).

The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect to the weight \(\left(1-x\right)^\alpha \left(1+x\right)^\beta\).

This functions returns the polynomials normilzed:

\[\int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n}\]

Examples

>>> from ... import jacobi_normalized
>>> from ...abc import n,a,b,x

>>> jacobi_normalized(n, a, b, x)
jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))

See also

gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Jacobi_polynomials

[2]

http://mathworld.wolfram.com/JacobiPolynomial.html

[3]

http://functions.wolfram.com/Polynomials/JacobiP/

class modelparameters.sympy.functions.special.polynomials.laguerre(n, x)

Bases: OrthogonalPolynomial

Returns the nth Laguerre polynomial in x, \(L_n(x)\).

Parameters:

n (int) – Degree of Laguerre polynomial. Must be n >= 0.

Examples

>>> from ... import laguerre, diff
>>> from ...abc import x, n
>>> laguerre(0, x)
1
>>> laguerre(1, x)
-x + 1
>>> laguerre(2, x)
x**2/2 - 2*x + 1
>>> laguerre(3, x)
-x**3/6 + 3*x**2/2 - 3*x + 1

>>> laguerre(n, x)
laguerre(n, x)

>>> diff(laguerre(n, x), x)
-assoc_laguerre(n - 1, 1, x)

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Laguerre_polynomial

[2]

http://mathworld.wolfram.com/LaguerrePolynomial.html

[3]

http://functions.wolfram.com/Polynomials/LaguerreL/

[4]

http://functions.wolfram.com/Polynomials/LaguerreL3/

default_assumptions = {}

classmethod eval(n, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.polynomials.legendre(n, x)

Bases: OrthogonalPolynomial

legendre(n, x) gives the nth Legendre polynomial of x, \(P_n(x)\)

The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy \(P_n(1) = 1\) for all n; further, \(P_n\) is odd for odd n and even for even n.

Examples

>>> from ... import legendre, diff
>>> from ...abc import x, n
>>> legendre(0, x)
1
>>> legendre(1, x)
x
>>> legendre(2, x)
3*x**2/2 - 1/2
>>> legendre(n, x)
legendre(n, x)
>>> diff(legendre(n,x), x)
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)

See also

jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly, sympy.polys.orthopolys.chebyshevt_poly, sympy.polys.orthopolys.chebyshevu_poly, sympy.polys.orthopolys.hermite_poly, sympy.polys.orthopolys.legendre_poly, sympy.polys.orthopolys.laguerre_poly

References

[1]

http://en.wikipedia.org/wiki/Legendre_polynomial

[2]

http://mathworld.wolfram.com/LegendrePolynomial.html

[3]

http://functions.wolfram.com/Polynomials/LegendreP/

[4]

http://functions.wolfram.com/Polynomials/LegendreP2/

default_assumptions = {}

classmethod eval(n, x)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=2)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.singularity_functions module

class modelparameters.sympy.functions.special.singularity_functions.SingularityFunction(variable, offset, exponent)

Bases: Function

The Singularity functions are a class of discontinuous functions. They take a variable, an offset and an exponent as arguments. These functions are represented using Macaulay brackets as :

SingularityFunction(x, a, n) := <x - a>^n

The singularity function will automatically evaluate to Derivative(DiracDelta(x - a), x, -n - 1) if n < 0 and (x - a)**n*Heaviside(x - a) if n >= 0.

Examples

>>> from ... import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol
>>> from ...abc import x, a, n
>>> SingularityFunction(x, a, n)
SingularityFunction(x, a, n)
>>> y = Symbol('y', positive=True)
>>> n = Symbol('n', nonnegative=True)
>>> SingularityFunction(y, -10, n)
(y + 10)**n
>>> y = Symbol('y', negative=True)
>>> SingularityFunction(y, 10, n)
0
>>> SingularityFunction(x, 4, -1).subs(x, 4)
oo
>>> SingularityFunction(x, 10, -2).subs(x, 10)
oo
>>> SingularityFunction(4, 1, 5)
243
>>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x)
4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)
>>> diff(SingularityFunction(x, 4, 0), x, 2)
SingularityFunction(x, 4, -2)
>>> SingularityFunction(x, 4, 5).rewrite(Piecewise)
Piecewise(((x - 4)**5, x - 4 > 0), (0, True))
>>> expr = SingularityFunction(x, a, n)
>>> y = Symbol('y', positive=True)
>>> n = Symbol('n', nonnegative=True)
>>> expr.subs({x: y, a: -10, n: n})
(y + 10)**n

The methods rewrite(DiracDelta), rewrite(Heaviside) and rewrite('HeavisideDiracDelta') returns the same output. One can use any of these methods according to their choice.

>>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2)
>>> expr.rewrite(Heaviside)
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
>>> expr.rewrite(DiracDelta)
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
>>> expr.rewrite('HeavisideDiracDelta')
(x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)

See also

DiracDelta, Heaviside

[1]

https://en.wikipedia.org/wiki/Singularity_function

default_assumptions = {'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'real': True}

classmethod eval(variable, offset, exponent)

Returns a simplified form or a value of Singularity Function depending on the argument passed by the object.

The eval() method is automatically called when the SingularityFunction class is about to be instantiated and it returns either some simplified instance or the unevaluated instance depending on the argument passed. In other words, eval() method is not needed to be called explicitly, it is being called and evaluated once the object is called.

Examples

>>> from ... import SingularityFunction, Symbol, nan
>>> from ...abc import x, a, n
>>> SingularityFunction(x, a, n)
SingularityFunction(x, a, n)
>>> SingularityFunction(5, 3, 2)
4
>>> SingularityFunction(x, a, nan)
nan
>>> SingularityFunction(x, 3, 0).subs(x, 3)
1
>>> SingularityFunction(x, a, n).eval(3, 5, 1)
0
>>> SingularityFunction(x, a, n).eval(4, 1, 5)
243
>>> x = Symbol('x', positive = True)
>>> a = Symbol('a', negative = True)
>>> n = Symbol('n', nonnegative = True)
>>> SingularityFunction(x, a, n)
(-a + x)**n
>>> x = Symbol('x', negative = True)
>>> a = Symbol('a', positive = True)
>>> SingularityFunction(x, a, n)
0

fdiff(argindex=1)

Returns the first derivative of a DiracDelta Function.

The difference between diff() and fdiff() is:- diff() is the user-level function and fdiff() is an object method. fdiff() is just a convenience method available in the Function class. It returns the derivative of the function without considering the chain rule. diff(function, x) calls Function._eval_derivative which in turn calls fdiff() internally to compute the derivative of the function.

is_commutative = True

is_complex = True

is_hermitian = True

is_imaginary = False

is_real = True

modelparameters.sympy.functions.special.spherical_harmonics module

class modelparameters.sympy.functions.special.spherical_harmonics.Ynm(n, m, theta, phi)

Bases: Function

Spherical harmonics defined as

\[Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}} \exp(i m \varphi) \mathrm{P}_n^m\left(\cos(\theta)\right)\]

Ynm() gives the spherical harmonic function of order n and m in theta and varphi, Y_n^m(theta, varphi). The four parameters are as follows: n geq 0 an integer and m an integer such that -n leq m leq n holds. The two angles are real-valued with theta in [0, pi] and varphi in [0, 2pi].

Examples

>>> from ... import Ynm, Symbol
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, m, theta, phi)
Ynm(n, m, theta, phi)

Several symmetries are known, for the order

>>> from ... import Ynm, Symbol
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, -m, theta, phi)
(-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

as well as for the angles

>>> from ... import Ynm, Symbol, simplify
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> Ynm(n, m, -theta, phi)
Ynm(n, m, theta, phi)

>>> Ynm(n, m, theta, -phi)
exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

For specific integers n and m we can evalute the harmonics to more useful expressions

>>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
1/(2*sqrt(pi))

>>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))

>>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
sqrt(3)*cos(theta)/(2*sqrt(pi))

>>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))

>>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))

>>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))

>>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))

>>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
-sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))

>>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))

We can differentiate the functions with respect to both angles

>>> from ... import Ynm, Symbol, diff
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> diff(Ynm(n, m, theta, phi), theta)
m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)

>>> diff(Ynm(n, m, theta, phi), phi)
I*m*Ynm(n, m, theta, phi)

Further we can compute the complex conjugation

>>> from ... import Ynm, Symbol, conjugate
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> conjugate(Ynm(n, m, theta, phi))
(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)

To get back the well known expressions in spherical coordinates we use full expansion

>>> from ... import Ynm, Symbol, expand_func
>>> from ...abc import n,m
>>> theta = Symbol("theta")
>>> phi = Symbol("phi")

>>> expand_func(Ynm(n, m, theta, phi))
sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))

See also

Ynm_c, Znm

References

[1]

http://en.wikipedia.org/wiki/Spherical_harmonics

[2]

http://mathworld.wolfram.com/SphericalHarmonic.html

[3]

http://functions.wolfram.com/Polynomials/SphericalHarmonicY/

[4]

http://dlmf.nist.gov/14.30

as_real_imag(deep=True, **hints)

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method can’t be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from .. import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from ..abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

default_assumptions = {}

classmethod eval(n, m, theta, phi)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=4)

Returns the first derivative of the function.

modelparameters.sympy.functions.special.spherical_harmonics.Ynm_c(n, m, theta, phi)

Conjugate spherical harmonics defined as

\[\overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi)\]

See also

Ynm, Znm

References

[1]

http://en.wikipedia.org/wiki/Spherical_harmonics

[2]

http://mathworld.wolfram.com/SphericalHarmonic.html

[3]

http://functions.wolfram.com/Polynomials/SphericalHarmonicY/

class modelparameters.sympy.functions.special.spherical_harmonics.Znm(n, m, theta, phi)

Bases: Function

Real spherical harmonics defined as

\[\begin{split}Z_n^m(\theta, \varphi) := \begin{cases} \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]

which gives in simplified form

\[\begin{split}Z_n^m(\theta, \varphi) = \begin{cases} \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\ Y_n^m(\theta, \varphi) &\quad m = 0 \\ \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\ \end{cases}\end{split}\]

See also

Ynm, Ynm_c

References

[1]

http://en.wikipedia.org/wiki/Spherical_harmonics

[2]

http://mathworld.wolfram.com/SphericalHarmonic.html

[3]

http://functions.wolfram.com/Polynomials/SphericalHarmonicY/

default_assumptions = {}

classmethod eval(n, m, theta, phi)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

modelparameters.sympy.functions.special.tensor_functions module

modelparameters.sympy.functions.special.tensor_functions.Eijk(*args, **kwargs)

Represent the Levi-Civita symbol.

This is just compatibility wrapper to LeviCivita().

See also

LeviCivita

class modelparameters.sympy.functions.special.tensor_functions.KroneckerDelta(i, j)

Bases: Function

The discrete, or Kronecker, delta function.

A function that takes in two integers i and j. It returns 0 if i and j are not equal or it returns 1 if i and j are equal.

Parameters:

• i (Number, Symbol) – The first index of the delta function.

• j (Number, Symbol) – The second index of the delta function.

Examples

A simple example with integer indices:

>>> from .tensor_functions import KroneckerDelta
>>> KroneckerDelta(1, 2)
0
>>> KroneckerDelta(3, 3)
1

Symbolic indices:

>>> from ...abc import i, j, k
>>> KroneckerDelta(i, j)
KroneckerDelta(i, j)
>>> KroneckerDelta(i, i)
1
>>> KroneckerDelta(i, i + 1)
0
>>> KroneckerDelta(i, i + 1 + k)
KroneckerDelta(i, i + k + 1)

See also

eval, sympy.functions.special.delta_functions.DiracDelta

References

[1]

http://en.wikipedia.org/wiki/Kronecker_delta

default_assumptions = {'algebraic': True, 'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'integer': True, 'irrational': False, 'noninteger': False, 'rational': True, 'real': True, 'transcendental': False}

classmethod eval(i, j)

Evaluates the discrete delta function.

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ...abc import i, j, k

>>> KroneckerDelta(i, j)
KroneckerDelta(i, j)
>>> KroneckerDelta(i, i)
1
>>> KroneckerDelta(i, i + 1)
0
>>> KroneckerDelta(i, i + 1 + k)
KroneckerDelta(i, i + k + 1)

# indirect doctest

property indices

property indices_contain_equal_information

Returns True if indices are either both above or below fermi.

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, q).indices_contain_equal_information
True
>>> KroneckerDelta(p, q+1).indices_contain_equal_information
True
>>> KroneckerDelta(i, p).indices_contain_equal_information
False

property is_above_fermi

True if Delta can be non-zero above fermi

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_above_fermi
True
>>> KroneckerDelta(p, i).is_above_fermi
False
>>> KroneckerDelta(p, q).is_above_fermi
True

See also

is_below_fermi, is_only_below_fermi, is_only_above_fermi

is_algebraic = True

property is_below_fermi

True if Delta can be non-zero below fermi

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_below_fermi
False
>>> KroneckerDelta(p, i).is_below_fermi
True
>>> KroneckerDelta(p, q).is_below_fermi
True

See also

is_above_fermi, is_only_above_fermi, is_only_below_fermi

is_commutative = True

is_complex = True

is_hermitian = True

is_imaginary = False

is_integer = True

is_irrational = False

is_noninteger = False

property is_only_above_fermi

True if Delta is restricted to above fermi

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, a).is_only_above_fermi
True
>>> KroneckerDelta(p, q).is_only_above_fermi
False
>>> KroneckerDelta(p, i).is_only_above_fermi
False

See also

is_above_fermi, is_below_fermi, is_only_below_fermi

property is_only_below_fermi

True if Delta is restricted to below fermi

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> q = Symbol('q')
>>> KroneckerDelta(p, i).is_only_below_fermi
True
>>> KroneckerDelta(p, q).is_only_below_fermi
False
>>> KroneckerDelta(p, a).is_only_below_fermi
False

See also

is_above_fermi, is_below_fermi, is_only_above_fermi

is_rational = True

is_real = True

is_transcendental = False

property killable_index

Returns the index which is preferred to substitute in the final expression.

The index to substitute is the index with less information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> j = Symbol('j', below_fermi=True)
>>> p = Symbol('p')
>>> KroneckerDelta(p, i).killable_index
p
>>> KroneckerDelta(p, a).killable_index
p
>>> KroneckerDelta(i, j).killable_index
j

See also

preferred_index

property preferred_index

Returns the index which is preferred to keep in the final expression.

The preferred index is the index with more information regarding fermi level. If indices contain same information, ‘a’ is preferred before ‘b’.

Examples

>>> from .tensor_functions import KroneckerDelta
>>> from ... import Symbol
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> j = Symbol('j', below_fermi=True)
>>> p = Symbol('p')
>>> KroneckerDelta(p, i).preferred_index
i
>>> KroneckerDelta(p, a).preferred_index
a
>>> KroneckerDelta(i, j).preferred_index
i

See also

killable_index

class modelparameters.sympy.functions.special.tensor_functions.LeviCivita(*args)

Bases: Function

Represent the Levi-Civita symbol.

For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0.

Thus it represents an alternating pseudotensor.

Examples

>>> from ... import LeviCivita
>>> from ...abc import i, j, k
>>> LeviCivita(1, 2, 3)
1
>>> LeviCivita(1, 3, 2)
-1
>>> LeviCivita(1, 2, 2)
0
>>> LeviCivita(i, j, k)
LeviCivita(i, j, k)
>>> LeviCivita(i, j, i)
0

See also

Eijk

default_assumptions = {'algebraic': True, 'commutative': True, 'complex': True, 'hermitian': True, 'imaginary': False, 'integer': True, 'irrational': False, 'noninteger': False, 'rational': True, 'real': True, 'transcendental': False}

doit()

Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.

>>> from .. import Integral
>>> from ..abc import x

>>> 2*Integral(x, x)
2*Integral(x, x)

>>> (2*Integral(x, x)).doit()
x**2

>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)

classmethod eval(*args)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

is_algebraic = True

is_commutative = True

is_complex = True

is_hermitian = True

is_imaginary = False

is_integer = True

is_irrational = False

is_noninteger = False

is_rational = True

is_real = True

is_transcendental = False

modelparameters.sympy.functions.special.tensor_functions.eval_levicivita(*args)

Evaluate Levi-Civita symbol.

modelparameters.sympy.functions.special.zeta_functions module

Riemann zeta and related function.

class modelparameters.sympy.functions.special.zeta_functions.dirichlet_eta(s)

Bases: Function

Dirichlet eta function.

For operatorname{Re}(s) > 0, this function is defined as

\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^n}{n^s}.\]

It admits a unique analytic continuation to all of \(\mathbb{C}\). It is an entire, unbranched function.

See also

zeta

References

[1]

http://en.wikipedia.org/wiki/Dirichlet_eta_function

Examples

The Dirichlet eta function is closely related to the Riemann zeta function:

>>> from ... import dirichlet_eta, zeta
>>> from ...abc import s
>>> dirichlet_eta(s).rewrite(zeta)
(-2**(-s + 1) + 1)*zeta(s)

default_assumptions = {}

classmethod eval(s)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.zeta_functions.lerchphi(*args)

Bases: Function

Lerch transcendent (Lerch phi function).

For \(\operatorname{Re}(a) > 0\), |z| < 1 and s in mathbb{C}, the Lerch transcendent is defined as

\[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\]

where the standard branch of the argument is used for \(n + a\), and by analytic continuation for other values of the parameters.

A commonly used related function is the Lerch zeta function, defined by

\[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\]

Analytic Continuation and Branching Behavior

It can be shown that

\[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\]

This provides the analytic continuation to operatorname{Re}(a) le 0.

Assume now operatorname{Re}(a) > 0. The integral representation

\[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} \frac{\mathrm{d}t}{\Gamma(s)}\]

provides an analytic continuation to \(\mathbb{C} - [1, \infty)\). Finally, for \(x \in (1, \infty)\) we find

\[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\]

using the standard branch for both \(\log{x}\) and \(\log{\log{x}}\) (a branch of \(\log{\log{x}}\) is needed to evaluate \(\log{x}^{s-1}\)). This concludes the analytic continuation. The Lerch transcendent is thus branched at \(z \in \{0, 1, \infty\}\) and \(a \in \mathbb{Z}_{\le 0}\). For fixed \(z, a\) outside these branch points, it is an entire function of \(s\).

See also

polylog, zeta

References

[1]

Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. Section 1.11.

[2]

http://dlmf.nist.gov/25.14

[3]

http://en.wikipedia.org/wiki/Lerch_transcendent

Examples

The Lerch transcendent is a fairly general function, for this reason it does not automatically evaluate to simpler functions. Use expand_func() to achieve this.

If \(z=1\), the Lerch transcendent reduces to the Hurwitz zeta function:

>>> from ... import lerchphi, expand_func
>>> from ...abc import z, s, a
>>> expand_func(lerchphi(1, s, a))
zeta(s, a)

More generally, if \(z\) is a root of unity, the Lerch transcendent reduces to a sum of Hurwitz zeta functions:

>>> expand_func(lerchphi(-1, s, a))
2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)

If \(a=1\), the Lerch transcendent reduces to the polylogarithm:

>>> expand_func(lerchphi(z, s, 1))
polylog(s, z)/z

More generally, if \(a\) is rational, the Lerch transcendent reduces to a sum of polylogarithms:

>>> from ... import S
>>> expand_func(lerchphi(z, s, S(1)/2))
2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
            polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))
>>> expand_func(lerchphi(z, s, S(3)/2))
-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
                      polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z

The derivatives with respect to \(z\) and \(a\) can be computed in closed form:

>>> lerchphi(z, s, a).diff(z)
(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z
>>> lerchphi(z, s, a).diff(a)
-s*lerchphi(z, s + 1, a)

default_assumptions = {}

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.zeta_functions.polylog(s, z)

Bases: Function

Polylogarithm function.

For \(|z| < 1\) and \(s \in \mathbb{C}\), the polylogarithm is defined by

\[\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},\]

where the standard branch of the argument is used for \(n\). It admits an analytic continuation which is branched at \(z=1\) (notably not on the sheet of initial definition), \(z=0\) and \(z=\infty\).

The name polylogarithm comes from the fact that for \(s=1\), the polylogarithm is related to the ordinary logarithm (see examples), and that

\[\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\]

The polylogarithm is a special case of the Lerch transcendent:

\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1)\]

See also

zeta, lerchphi

Examples

For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed using other functions:

>>> from ... import polylog
>>> from ...abc import s
>>> polylog(s, 0)
0
>>> polylog(s, 1)
zeta(s)
>>> polylog(s, -1)
-dirichlet_eta(s)

If \(s\) is a negative integer, \(0\) or \(1\), the polylogarithm can be expressed using elementary functions. This can be done using expand_func():

>>> from ... import expand_func
>>> from ...abc import z
>>> expand_func(polylog(1, z))
-log(z*exp_polar(-I*pi) + 1)
>>> expand_func(polylog(0, z))
z/(-z + 1)

The derivative with respect to \(z\) can be computed in closed form:

>>> polylog(s, z).diff(z)
polylog(s - 1, z)/z

The polylogarithm can be expressed in terms of the lerch transcendent:

>>> from ... import lerchphi
>>> polylog(s, z).rewrite(lerchphi)
z*lerchphi(z, s, 1)

default_assumptions = {}

classmethod eval(s, z)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

class modelparameters.sympy.functions.special.zeta_functions.stieltjes(n, a=None)

Bases: Function

Represents Stieltjes constants, \(\gamma_{k}\) that occur in Laurent Series expansion of the Riemann zeta function.

Examples

>>> from ... import stieltjes
>>> from ...abc import n, m
>>> stieltjes(n)
stieltjes(n)

zero’th stieltjes constant

>>> stieltjes(0)
EulerGamma
>>> stieltjes(0, 1)
EulerGamma

For generalized stieltjes constants

>>> stieltjes(n, m)
stieltjes(n, m)

Constants are only defined for integers >= 0

>>> stieltjes(-1)
zoo

References

[1]

http://en.wikipedia.org/wiki/Stieltjes_constants

default_assumptions = {}

classmethod eval(n, a=None)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

class modelparameters.sympy.functions.special.zeta_functions.zeta(z, a_=None)

Bases: Function

Hurwitz zeta function (or Riemann zeta function).

For operatorname{Re}(a) > 0 and operatorname{Re}(s) > 1, this function is defined as

\[\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},\]

where the standard choice of argument for \(n + a\) is used. For fixed \(a\) with operatorname{Re}(a) > 0 the Hurwitz zeta function admits a meromorphic continuation to all of \(\mathbb{C}\), it is an unbranched function with a simple pole at \(s = 1\).

Analytic continuation to other \(a\) is possible under some circumstances, but this is not typically done.

The Hurwitz zeta function is a special case of the Lerch transcendent:

\[\zeta(s, a) = \Phi(1, s, a).\]

This formula defines an analytic continuation for all possible values of \(s\) and \(a\) (also operatorname{Re}(a) < 0), see the documentation of lerchphi for a description of the branching behavior.

If no value is passed for \(a\), by this function assumes a default value of \(a = 1\), yielding the Riemann zeta function.

See also

dirichlet_eta, lerchphi, polylog

References

[1]

http://dlmf.nist.gov/25.11

[2]

http://en.wikipedia.org/wiki/Hurwitz_zeta_function

Examples

For \(a = 1\) the Hurwitz zeta function reduces to the famous Riemann zeta function:

\[\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\]

>>> from ... import zeta
>>> from ...abc import s
>>> zeta(s, 1)
zeta(s)
>>> zeta(s)
zeta(s)

The Riemann zeta function can also be expressed using the Dirichlet eta function:

>>> from ... import dirichlet_eta
>>> zeta(s).rewrite(dirichlet_eta)
dirichlet_eta(s)/(-2**(-s + 1) + 1)

The Riemann zeta function at positive even integer and negative odd integer values is related to the Bernoulli numbers:

>>> zeta(2)
pi**2/6
>>> zeta(4)
pi**4/90
>>> zeta(-1)
-1/12

The specific formulae are:

\[\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}\]

\[\zeta(-n) = -\frac{B_{n+1}}{n+1}\]

At negative even integers the Riemann zeta function is zero:

>>> zeta(-4)
0

No closed-form expressions are known at positive odd integers, but numerical evaluation is possible:

>>> zeta(3).n()
1.20205690315959

The derivative of \(\zeta(s, a)\) with respect to \(a\) is easily computed:

>>> from ...abc import a
>>> zeta(s, a).diff(a)
-s*zeta(s + 1, a)

However the derivative with respect to \(s\) has no useful closed form expression:

>>> zeta(s, a).diff(s)
Derivative(zeta(s, a), s)

The Hurwitz zeta function can be expressed in terms of the Lerch transcendent, sympy.functions.special.lerchphi:

>>> from ... import lerchphi
>>> zeta(s, a).rewrite(lerchphi)
lerchphi(1, s, a)

default_assumptions = {}

classmethod eval(z, a_=None)

Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be instantiated and it should return either some simplified instance (possible of some other class), or if the class cls should be unmodified, return None.

Examples of eval() for the function “sign”

@classmethod def eval(cls, arg):

if arg is S.NaN:

return S.NaN

if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Mul):

coeff, terms = arg.as_coeff_Mul(rational=True) if coeff is not S.One:

return cls(coeff) * cls(terms)

fdiff(argindex=1)

Returns the first derivative of the function.

Module contents