Special #

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

[R343]

[R344]

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

inverse(argindex=1)[source]#: Returns the inverse of this function.

class sympy.functions.special.error_functions.erfc(arg)[source]#

Complementary Error Function.

Explanation

The function is defined as:

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

Examples

>>> from sympy import I, oo, erfc
>>> from sympy.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 sympy import conjugate
>>> conjugate(erfc(z))
erfc(conjugate(z))

Differentiation with respect to \(z\) is supported:

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

It also follows

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

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

[R345]

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

[R346]

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

[R347]

[R348]

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

inverse(argindex=1)[source]#: Returns the inverse of this function.

class sympy.functions.special.error_functions.erfi(z)[source]#

Imaginary error function.

Explanation

The function erfi is defined as:

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

Examples

>>> from sympy import I, oo, erfi
>>> from sympy.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 sympy import conjugate
>>> conjugate(erfi(z))
erfi(conjugate(z))

Differentiation with respect to \(z\) is supported:

>>> from sympy 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

[R349]

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

[R350]

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

[R351]

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

class sympy.functions.special.error_functions.erf2(x, y)[source]#

Two-argument error function.

Explanation

This function is defined as:

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

Examples

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

Several special values are known:

>>> erf2(0, 0)
0
>>> erf2(x, x)
0
>>> erf2(x, oo)
1 - erf(x)
>>> 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 sympy import conjugate
>>> conjugate(erf2(x, y))
erf2(conjugate(x), conjugate(y))

Differentiation with respect to \(x\), \(y\) is supported:

>>> from sympy 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

[R352]

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

class sympy.functions.special.error_functions.erfinv(z)[source]#

Inverse Error Function. The erfinv function is defined as:

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

Examples

>>> from sympy import erfinv
>>> from sympy.abc import x

Several special values are known:

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

Differentiation with respect to \(x\) is supported:

>>> from sympy 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

[R353]

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

[R354]

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

inverse(argindex=1)[source]#: Returns the inverse of this function.

class sympy.functions.special.error_functions.erfcinv(z)[source]#

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

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

Examples

>>> from sympy import erfcinv
>>> from sympy.abc import x

Several special values are known:

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

Differentiation with respect to \(x\) is supported:

>>> from sympy 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

[R355]

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

[R356]

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

inverse(argindex=1)[source]#: Returns the inverse of this function.

class sympy.functions.special.error_functions.erf2inv(x, y)[source]#

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 sympy import erf2inv, oo
>>> from sympy.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 sympy 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

[R357]

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

class sympy.functions.special.error_functions.FresnelIntegral(z)[source]#: Base class for the Fresnel integrals.

class sympy.functions.special.error_functions.fresnels(z)[source]#

Fresnel integral S.

Explanation

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 sympy import I, oo, fresnels
>>> from sympy.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 sympy import conjugate
>>> conjugate(fresnels(z))
fresnels(conjugate(z))

Differentiation with respect to \(z\) is supported:

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

Defining the Fresnel functions via an integral:

>>> from sympy import integrate, pi, sin, 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

[R358]

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

[R359]

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

[R360]

[R361]

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

[R362]

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

class sympy.functions.special.error_functions.fresnelc(z)[source]#

Fresnel integral C.

Explanation

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 sympy import I, oo, fresnelc
>>> from sympy.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 sympy import conjugate
>>> conjugate(fresnelc(z))
fresnelc(conjugate(z))

Differentiation with respect to \(z\) is supported:

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

Defining the Fresnel functions via an integral:

>>> from sympy import integrate, pi, cos, 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

[R363]

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

[R364]

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

[R365]

[R366]

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

[R367]

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

Exponential, Logarithmic and Trigonometric Integrals#

class sympy.functions.special.error_functions.Ei(z)[source]#

The classical exponential integral.

Explanation

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.

Examples

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

>>> Ei(-1)
Ei(-1)

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 sympy import 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.
uppergamma: Upper incomplete gamma function.

References

[R368]

http://dlmf.nist.gov/6.6

[R369]

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

[R370]

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

class sympy.functions.special.error_functions.expint(nu, z)[source]#

Generalized exponential integral.

Explanation

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

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

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

Differentiation is supported. Differentiation with respect to \(z\) further explains 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, 1 - nu), ()), 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 sympy 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 sympy import uppergamma
>>> expint(nu, z).rewrite(uppergamma)
z**(nu - 1)*uppergamma(1 - nu, z)

As such it is branched at the origin:

>>> from sympy 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(1 - nu) + 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.

uppergamma

References

[R371]

http://dlmf.nist.gov/8.19

[R372]

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

[R373]

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

sympy.functions.special.error_functions.E1(z)[source]#

Classical case of the generalized exponential integral.

Explanation

This is equivalent to expint(1, z).

Examples

>>> from sympy import E1
>>> E1(0)
expint(1, 0)

>>> E1(5)
expint(1, 5)

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 sympy.functions.special.error_functions.li(z)[source]#

The classical logarithmic integral.

Explanation

For use in SymPy, this function is defined as

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

Examples

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

Defining the li function via an integral: >>> from sympy import integrate >>> integrate(li(z)) z*li(z) - Ei(2*log(z))

>>> integrate(li(z),z)
z*li(z) - Ei(2*log(z))

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

>>> from sympy 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 sympy 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

[R374]

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

[R375]

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

[R376]

http://dlmf.nist.gov/6

[R377]

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

class sympy.functions.special.error_functions.Li(z)[source]#

The offset logarithmic integral.

Explanation

For use in SymPy, this function is defined as

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

Examples

>>> from sympy import Li
>>> from sympy.abc import z

The following special value is known:

>>> Li(2)
0

Differentiation with respect to \(z\) is supported:

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

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

>>> from sympy 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

[R378]

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

[R379]

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

[R380]

http://dlmf.nist.gov/6

class sympy.functions.special.error_functions.Si(z)[source]#

Sine integral.

Explanation

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

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

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

It is unbranched:

>>> from sympy 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 sympy 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 sympy 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

[R381]

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

class sympy.functions.special.error_functions.Ci(z)[source]#

Cosine integral.

Explanation

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 sympy import Ci
>>> from sympy.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 sympy 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 sympy 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 sympy 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

[R382]

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

class sympy.functions.special.error_functions.Shi(z)[source]#

Sinh integral.

Explanation

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

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

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

It is unbranched:

>>> from sympy 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 sympy 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

[R383]

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

class sympy.functions.special.error_functions.Chi(z)[source]#

Cosh integral.

Explanation

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 sympy import Chi
>>> from sympy.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 sympy 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 sympy 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 sympy 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

[R384]

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

Bessel Type Functions#

class sympy.functions.special.bessel.BesselBase(nu, z)[source]#

Abstract base class for Bessel-type functions.

This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with 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.

property order#: The order of the Bessel-type function.

class sympy.functions.special.bessel.besselj(nu, z)[source]#

Bessel function of the first kind.

Explanation

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 sympy import besselj, jn
>>> from sympy.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

[R385]

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

[R386]

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

[R387]

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

[R388]

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

class sympy.functions.special.bessel.bessely(nu, z)[source]#

Bessel function of the second kind.

Explanation

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 sympy import bessely, yn
>>> from sympy.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

[R389]

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

class sympy.functions.special.bessel.besseli(nu, z)[source]#

Modified Bessel function of the first kind.

Explanation

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 sympy import besseli
>>> from sympy.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

[R390]

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

class sympy.functions.special.bessel.besselk(nu, z)[source]#

Modified Bessel function of the second kind.

Explanation

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 sympy import besselk
>>> from sympy.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

[R391]

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

class sympy.functions.special.bessel.hankel1(nu, z)[source]#

Hankel function of the first kind.

Explanation

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 sympy import hankel1
>>> from sympy.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

[R392]

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

class sympy.functions.special.bessel.hankel2(nu, z)[source]#

Hankel function of the second kind.

Explanation

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 sympy import hankel2
>>> from sympy.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

[R393]

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

class sympy.functions.special.bessel.jn(nu, z)[source]#

Spherical Bessel function of the first kind.

Explanation

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 sympy.polys.orthopolys.spherical_bessel_fn().

Examples

>>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely
>>> 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

[R394]

http://dlmf.nist.gov/10.47

class sympy.functions.special.bessel.yn(nu, z)[source]#

Spherical Bessel function of the second kind.

Explanation

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 sympy 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

[R395]

http://dlmf.nist.gov/10.47

sympy.functions.special.bessel.jn_zeros(n, k, method='sympy', dps=15)[source]#

Zeros of the spherical Bessel function of the first kind.

Parameters:

n : integer

order of Bessel function

k : integer

number of zeros to return

Explanation

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

class sympy.functions.special.bessel.marcumq(m, a, b)[source]#

The Marcum Q-function.

Explanation

The Marcum Q-function is defined by the meromorphic continuation of

\[Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx\]

Examples

>>> from sympy import marcumq
>>> from sympy.abc import m, a, b
>>> marcumq(m, a, b)
marcumq(m, a, b)

Special values:

>>> marcumq(m, 0, b)
uppergamma(m, b**2/2)/gamma(m)
>>> marcumq(0, 0, 0)
0
>>> marcumq(0, a, 0)
1 - exp(-a**2/2)
>>> marcumq(1, a, a)
1/2 + exp(-a**2)*besseli(0, a**2)/2
>>> marcumq(2, a, a)
1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)

Differentiation with respect to \(a\) and \(b\) is supported:

>>> from sympy import diff
>>> diff(marcumq(m, a, b), a)
a*(-marcumq(m, a, b) + marcumq(m + 1, a, b))
>>> diff(marcumq(m, a, b), b)
-a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b)

References

[R396]

https://en.wikipedia.org/wiki/Marcum_Q-function

[R397]

http://mathworld.wolfram.com/MarcumQ-Function.html

Airy Functions#

class sympy.functions.special.bessel.AiryBase(*args)[source]#

Abstract base class for Airy functions.

This class is meant to reduce code duplication.

class sympy.functions.special.bessel.airyai(arg)[source]#

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

Explanation

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

>>> airyai(z)
airyai(z)

Several special values are known:

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

The Airy function obeys the mirror symmetry:

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

Differentiation with respect to \(z\) is supported:

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

Series expansion is also supported:

>>> from sympy 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 \(\operatorname{Ai}(z)\) in terms of hypergeometric functions:

>>> from sympy 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

[R398]

[R399]

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

[R400]

[R401]

class sympy.functions.special.bessel.airybi(arg)[source]#

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

Explanation

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

>>> airybi(z)
airybi(z)

Several special values are known:

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

The Airy function obeys the mirror symmetry:

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

Differentiation with respect to \(z\) is supported:

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

Series expansion is also supported:

>>> from sympy 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 \(\operatorname{Bi}(z)\) in terms of hypergeometric functions:

>>> from sympy 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

[R402]

[R403]

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

[R404]

[R405]

class sympy.functions.special.bessel.airyaiprime(arg)[source]#

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

Explanation

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

>>> airyaiprime(z)
airyaiprime(z)

Several special values are known:

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

The Airy function obeys the mirror symmetry:

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

Differentiation with respect to \(z\) is supported:

>>> from sympy 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 sympy 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 \(\operatorname{Ai}^\prime(z)\) in terms of hypergeometric functions:

>>> from sympy 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

[R406]

[R407]

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

[R408]

[R409]

class sympy.functions.special.bessel.airybiprime(arg)[source]#

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

Explanation

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

>>> airybiprime(z)
airybiprime(z)

Several special values are known:

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

The Airy function obeys the mirror symmetry:

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

Differentiation with respect to \(z\) is supported:

>>> from sympy 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 sympy 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 \(\operatorname{Bi}^\prime(z)\) in terms of hypergeometric functions:

>>> from sympy 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

[R410]

[R411]

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

[R412]

[R413]

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

B-Splines#

sympy.functions.special.bsplines.bspline_basis(d, knots, n, x)[source]#

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

Parameters:

d : integer

degree of bspline

knots : list of integer values

list of knots points of bspline

n : integer

\(n\)-th B-spline

x : symbol

Explanation

B-Splines are piecewise polynomials of degree \(d\). They are defined on a set of knots, which is a sequence of integers or floats.

Examples

The 0th degree splines have a value of 1 on a single interval:

>>> from sympy import bspline_basis
>>> from sympy.abc import x
>>> d = 0
>>> knots = tuple(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, tuple(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((1 - x/2, (x >= 0) & (x <= 2)), (0, True))

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

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

See also

bspline_basis_set

References

[R414]

sympy.functions.special.bsplines.bspline_basis_set(d, knots, x)[source]#

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

Parameters:

d : integer

degree of bspline

knots : list of integers

list of knots points of bspline

x : symbol

Explanation

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 sympy import bspline_basis_set
>>> from sympy.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

bspline_basis

sympy.functions.special.bsplines.interpolating_spline(d, x, X, Y)[source]#

Return spline of degree d, passing through the given X and Y values.

Parameters:

d : integer

Degree of Bspline strictly greater than equal to one

x : symbol

X : list of strictly increasing integer values

list of X coordinates through which the spline passes

Y : list of strictly increasing integer values

list of Y coordinates through which the spline passes

Explanation

This function returns a piecewise function such that each part is a polynomial of degree not greater than d. The value of d must be 1 or greater and the values of X must be strictly increasing.

Examples

>>> from sympy import interpolating_spline
>>> from sympy.abc import x
>>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
Piecewise((3*x, (x >= 1) & (x <= 2)),
        (7 - x/2, (x >= 2) & (x <= 4)),
        (2*x/3 + 7/3, (x >= 4) & (x <= 7)))
>>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)),
        (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))

Riemann Zeta and Related Functions#

class sympy.functions.special.zeta_functions.zeta(z, a_=None)[source]#

Hurwitz zeta function (or Riemann zeta function).

Explanation

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.

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 sympy import zeta
>>> from sympy.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 sympy import dirichlet_eta
>>> zeta(s).rewrite(dirichlet_eta)
dirichlet_eta(s)/(1 - 2**(1 - s))

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 \(a\) can be computed:

>>> from sympy.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, lerchphi:

>>> from sympy import lerchphi
>>> zeta(s, a).rewrite(lerchphi)
lerchphi(1, s, a)

See also

dirichlet_eta, lerchphi, polylog

References

[R415]

http://dlmf.nist.gov/25.11

[R416]

https://en.wikipedia.org/wiki/Hurwitz_zeta_function

class sympy.functions.special.zeta_functions.dirichlet_eta(s)[source]#

Dirichlet eta function.

Explanation

For \(\operatorname{Re}(s) > 0\), this function is defined as

\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.\]

It admits a unique analytic continuation to all of \(\mathbb{C}\). It is an entire, unbranched function.

Examples

The Dirichlet eta function is closely related to the Riemann zeta function:

>>> from sympy import dirichlet_eta, zeta
>>> from sympy.abc import s
>>> dirichlet_eta(s).rewrite(zeta)
(1 - 2**(1 - s))*zeta(s)

See also

zeta

References

[R417]

https://en.wikipedia.org/wiki/Dirichlet_eta_function

class sympy.functions.special.zeta_functions.polylog(s, z)[source]#

Polylogarithm function.

Explanation

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:

\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1).\]

Examples

For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed using other functions:

>>> from sympy import polylog
>>> from sympy.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 sympy import expand_func
>>> from sympy.abc import z
>>> expand_func(polylog(1, z))
-log(1 - z)
>>> expand_func(polylog(0, z))
z/(1 - z)

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 sympy import lerchphi
>>> polylog(s, z).rewrite(lerchphi)
z*lerchphi(z, s, 1)

See also

zeta, lerchphi

class sympy.functions.special.zeta_functions.lerchphi(*args)[source]#

Lerch transcendent (Lerch phi function).

Explanation

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 \(s\).

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 sympy import lerchphi, expand_func
>>> from sympy.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))
zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s

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 sympy 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)

See also

polylog, zeta

References

[R418]

Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. Section 1.11.

[R419]

http://dlmf.nist.gov/25.14

[R420]

https://en.wikipedia.org/wiki/Lerch_transcendent

class sympy.functions.special.zeta_functions.stieltjes(n, a=None)[source]#

Represents Stieltjes constants, \(\gamma_{k}\) that occur in Laurent Series expansion of the Riemann zeta function.

Examples

>>> from sympy import stieltjes
>>> from sympy.abc import n, m
>>> stieltjes(n)
stieltjes(n)

The 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

[R421]

https://en.wikipedia.org/wiki/Stieltjes_constants

Hypergeometric Functions#

class sympy.functions.special.hyper.hyper(ap, bq, z)[source]#

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.

Explanation

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 \(z\).

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

You can also use expand_func():

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

More examples:

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

References

[R422]

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

[R423]

https://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.

property eta#: A quantity related to the convergence of the series.

property radius_of_convergence#

Compute the radius of convergence of the defining series.

Explanation

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.

Examples

>>> from sympy import hyper
>>> from sympy.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 sympy.functions.special.hyper.meijerg(*args)[source]#

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

Explanation

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.

Please 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 sympy import meijerg, Tuple, pprint
>>> from sympy.abc import x, a
>>> 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 sympy 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 sympy import hyperexpand
>>> from sympy.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 sympy 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

[R424]

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

[R425]

https://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.

property delta#: A quantity related to the convergence region of the integral, c.f. references.

get_period()[source]#

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

Examples

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

>>> 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)[source]#: Get the defining integrand D(s).

property is_number#: Returns true if expression has numeric data only.

property nu#: A quantity related to the convergence region of the integral, c.f. references.

class sympy.functions.special.hyper.appellf1(a, b1, b2, c, x, y)[source]#

This is the Appell hypergeometric function of two variables as:

\[F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}} \frac{x^m y^n}{m! n!}.\]

Examples

>>> from sympy import appellf1, symbols
>>> x, y, a, b1, b2, c = symbols('x y a b1 b2 c')
>>> appellf1(2., 1., 6., 4., 5., 6.)
0.0063339426292673
>>> appellf1(12., 12., 6., 4., 0.5, 0.12)
172870711.659936
>>> appellf1(40, 2, 6, 4, 15, 60)
appellf1(40, 2, 6, 4, 15, 60)
>>> appellf1(20., 12., 10., 3., 0.5, 0.12)
15605338197184.4
>>> appellf1(40, 2, 6, 4, x, y)
appellf1(40, 2, 6, 4, x, y)
>>> appellf1(a, b1, b2, c, x, y)
appellf1(a, b1, b2, c, x, y)

References

[R426]

https://en.wikipedia.org/wiki/Appell_series

[R427]

http://functions.wolfram.com/HypergeometricFunctions/AppellF1/

Elliptic integrals#

class sympy.functions.special.elliptic_integrals.elliptic_k(m)[source]#

The complete elliptic integral of the first kind, defined by

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

where \(F\left(z\middle| m\right)\) is the Legendre incomplete elliptic integral of the first kind.

Explanation

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 sympy import elliptic_k, I
>>> from sympy.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)

See also

elliptic_f

References

[R428]

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

[R429]

class sympy.functions.special.elliptic_integrals.elliptic_f(z, m)[source]#

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

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

Explanation

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

Examples

>>> from sympy import elliptic_f, I
>>> from sympy.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

See also

elliptic_k

References

[R430]

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

[R431]

class sympy.functions.special.elliptic_integrals.elliptic_e(m, z=None)[source]#

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

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

Explanation

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

Examples

>>> from sympy import elliptic_e, I
>>> from sympy.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

[R432]

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

[R433]

[R434]

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

class sympy.functions.special.elliptic_integrals.elliptic_pi(n, m, z=None)[source]#

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:

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

Explanation

Examples

>>> from sympy import elliptic_pi, I
>>> from sympy.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

[R435]

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

[R436]

[R437]

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

Mathieu Functions#

class sympy.functions.special.mathieu_functions.MathieuBase(*args)[source]#

Abstract base class for Mathieu functions.

This class is meant to reduce code duplication.

class sympy.functions.special.mathieu_functions.mathieus(a, q, z)[source]#

The Mathieu Sine function \(S(a,q,z)\).

Explanation

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 sympy import diff, mathieus
>>> from sympy.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

[R438]

[R439]

[R440]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/

[R441]

class sympy.functions.special.mathieu_functions.mathieuc(a, q, z)[source]#

The Mathieu Cosine function \(C(a,q,z)\).

Explanation

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 sympy import diff, mathieuc
>>> from sympy.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

[R442]

[R443]

[R444]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/

[R445]

class sympy.functions.special.mathieu_functions.mathieusprime(a, q, z)[source]#

The derivative \(S^{\prime}(a,q,z)\) of the Mathieu Sine function.

Explanation

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 sympy import diff, mathieusprime
>>> from sympy.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

[R446]

[R447]

[R448]

http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/

[R449]

class sympy.functions.special.mathieu_functions.mathieucprime(a, q, z)[source]#

The derivative \(C^{\prime}(a,q,z)\) of the Mathieu Cosine function.

Explanation

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 sympy import diff, mathieucprime
>>> from sympy.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

[R450]

[R451]

[R452]