# Special Functions¶

AUTHORS:

• David Joyner (2006-13-06): initial version
• David Joyner (2006-30-10): bug fixes to pari wrappers of Bessel functions, hypergeometric_U
• William Stein (2008-02): Impose some sanity checks.
• David Joyner (2008-04-23): addition of elliptic integrals

This module provides easy access to many of Maxima and PARI’s special functions.

Maxima’s special functions package (which includes spherical harmonic functions, spherical Bessel functions (of the 1st and 2nd kind), and spherical Hankel functions (of the 1st and 2nd kind)) was written by Barton Willis of the University of Nebraska at Kearney. It is released under the terms of the General Public License (GPL).

Support for elliptic functions and integrals was written by Raymond Toy. It is placed under the terms of the General Public License (GPL) that governs the distribution of Maxima.

Next, we summarize some of the properties of the functions implemented here.

• Airy function The function $$Ai(x)$$ and the related function $$Bi(x)$$, which is also called an Airy function, are solutions to the differential equation

$y'' - xy = 0,$

known as the Airy equation. They belong to the class of ‘Bessel functions of fractional order’. The initial conditions $$Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}$$, $$Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}$$ define $$Ai(x)$$. The initial conditions $$Bi(0) = 3^{1/2}Ai(0)$$, $$Bi'(0) = -3^{1/2}Ai'(0)$$ define $$Bi(x)$$.

They are named after the British astronomer George Biddell Airy.

• Spherical harmonics: Laplace’s equation in spherical coordinates is:

${\frac{1}{r^2}}{\frac{\partial}{\partial r}} \left(r^2 {\frac{\partial f}{\partial r}}\right) + {\frac{1}{r^2}\sin\theta}{\frac{\partial}{\partial \theta}} \left(\sin\theta {\frac{\partial f}{\partial \theta}}\right) + {\frac{1}{r^2\sin^2\theta}}{\frac{\partial^2 f}{\partial \varphi^2}} = 0.$

Note that the spherical coordinates $$\theta$$ and $$\varphi$$ are defined here as follows: $$\theta$$ is the colatitude or polar angle, ranging from $$0\leq\theta\leq\pi$$ and $$\varphi$$ the azimuth or longitude, ranging from $$0\leq\varphi<2\pi$$.

The general solution which remains finite towards infinity is a linear combination of functions of the form

$r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} )$

and

$r^{-1-\ell} \sin (m \varphi) P_\ell^m (\cos{\theta} )$

where $$P_\ell^m$$ are the associated Legendre polynomials, and with integer parameters $$\ell \ge 0$$ and $$m$$ from $$0$$ to $$\ell$$. Put in another way, the solutions with integer parameters $$\ell \ge 0$$ and $$- \ell\leq m\leq \ell$$, can be written as linear combinations of:

$U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )$

where the functions $$Y$$ are the spherical harmonic functions with parameters $$\ell$$, $$m$$, which can be written as:

$Y_\ell^m( \theta , \varphi ) = \sqrt{{\frac{(2\ell+1)}{4\pi}}{\frac{(\ell-m)!}{(\ell+m)!}}} \cdot e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) .$

The spherical harmonics obey the normalisation condition

$\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi} Y_\ell^mY_{\ell'}^{m'*}\,d\Omega =\delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega =\sin\theta\,d\varphi\,d\theta .$
• When solving for separable solutions of Laplace’s equation in spherical coordinates, the radial equation has the form:

$x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.$

The spherical Bessel functions $$j_n$$ and $$y_n$$, are two linearly independent solutions to this equation. They are related to the ordinary Bessel functions $$J_n$$ and $$Y_n$$ by:

$j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),$
$y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).$
• For $$x>0$$, the confluent hypergeometric function $$y = U(a,b,x)$$ is defined to be the solution to Kummer’s differential equation

$xy'' + (b-x)y' - ay = 0,$

which satisfies $$U(a,b,x) \sim x^{-a}$$, as $$x\rightarrow \infty$$. (There is a linearly independent solution, called Kummer’s function $$M(a,b,x)$$, which is not implemented.)

• The incomplete elliptic integrals (of the first kind, etc.) are:

$\begin{split}\begin{array}{c} \displaystyle\int_0^\phi \frac{1}{\sqrt{1 - m\sin(x)^2}}\, dx,\\ \displaystyle\int_0^\phi \sqrt{1 - m\sin(x)^2}\, dx,\\ \displaystyle\int_0^\phi \frac{\sqrt{1-mt^2}}{\sqrt(1 - t^2)}\, dx,\\ \displaystyle\int_0^\phi \frac{1}{\sqrt{1 - m\sin(x)^2\sqrt{1 - n\sin(x)^2}}}\, dx, \end{array}\end{split}$

and the complete ones are obtained by taking $$\phi =\pi/2$$.

REFERENCES:

TODO: Resolve weird bug in commented out code in hypergeometric_U below.

AUTHORS:

• David Joyner and William Stein

Added 16-02-2008 (wdj): optional calls to scipy and replace all ‘#random’ by ‘...’ (both at the request of William Stein)

Warning

SciPy’s versions are poorly documented and seem less accurate than the Maxima and PARI versions; typically they are limited by hardware floats precision.

class sage.functions.special.EllipticE

This returns the value of the “incomplete elliptic integral of the second kind,”

$\int_0^\phi \sqrt{1 - m\sin(x)^2}\, dx,$

i.e., integrate(sqrt(1 - m*sin(x)^2), x, 0, phi). Taking $$\phi = \pi/2$$ gives elliptic_ec.

EXAMPLES:

sage: z = var("z")
sage: # this is still wrong: must be abs(sin(z)) + 2*round(z/pi)
sage: elliptic_e(z, 1)
2*round(z/pi) + sin(z)
sage: elliptic_e(z, 0)
z
sage: elliptic_e(0.5, 0.1)  # abs tol 2e-15
0.498011394498832

class sage.functions.special.EllipticEC

This returns the value of the “complete elliptic integral of the second kind,”

$\int_0^{\pi/2} \sqrt{1 - m\sin(x)^2}\, dx.$

EXAMPLES:

sage: elliptic_ec(0.1)
1.53075763689776
sage: elliptic_ec(x).diff()
1/2*(elliptic_ec(x) - elliptic_kc(x))/x

elliptic_ec

class sage.functions.special.EllipticEU

This returns the value of the “incomplete elliptic integral of the second kind,”

$\int_0^u \mathrm{dn}(x,m)^2\, dx = \int_0^\tau {\sqrt{1-m x^2}\over\sqrt{1-x^2}}\, dx.$

where $$\tau = \mathrm{sn}(u, m)$$.

EXAMPLES:

sage: elliptic_eu (0.5, 0.1)
0.496054551286597

class sage.functions.special.EllipticF

This returns the value of the “incomplete elliptic integral of the first kind,”

$\int_0^\phi \frac{dx}{\sqrt{1 - m\sin(x)^2}},$

i.e., integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi). Taking $$\phi = \pi/2$$ gives elliptic_kc.

EXAMPLES:

sage: z = var("z")
sage: elliptic_f (z, 0)
z
sage: elliptic_f (z, 1)
log(tan(1/4*pi + 1/2*z))
sage: elliptic_f (0.2, 0.1)
0.200132506747543

class sage.functions.special.EllipticKC

This returns the value of the “complete elliptic integral of the first kind,”

$\int_0^{\pi/2} \frac{dx}{\sqrt{1 - m\sin(x)^2}}.$

EXAMPLES:

sage: elliptic_kc(0.5)
1.85407467730137
sage: elliptic_f(RR(pi/2), 0.5)
1.85407467730137

class sage.functions.special.EllipticPi

This returns the value of the “incomplete elliptic integral of the third kind,”

$\text{elliptic\_pi}(n, t, m) = \int_0^t \frac{dx}{(1 - n \sin(x)^2) \sqrt{1 - m \sin(x)^2}}.$

INPUT:

• n – a real number, called the “characteristic”
• t – a real number, called the “amplitude”
• m – a real number, called the “parameter”

EXAMPLES:

sage: N(elliptic_pi(1, pi/4, 1))
1.14779357469632


Compare the value computed by Maxima to the definition as a definite integral (using GSL):

sage: elliptic_pi(0.1, 0.2, 0.3)
0.200665068220979
sage: numerical_integral(1/(1-0.1*sin(x)^2)/sqrt(1-0.3*sin(x)^2), 0.0, 0.2)
(0.2006650682209791, 2.227829789769088e-15)


ALGORITHM:

Numerical evaluation and symbolic manipulation are provided by Maxima.

REFERENCES:

class sage.functions.special.MaximaFunction(name, nargs=2, conversions={})

EXAMPLES:

sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f(1,1)
tanh(1)
sage: f(1/2,1/2).n()
0.470750473655657

class sage.functions.special.SphericalHarmonic

Returns the spherical harmonic function $$Y_n^m(\theta, \varphi)$$.

For integers $$n > -1$$, $$|m| \leq n$$, simplification is done automatically. Numeric evaluation is supported for complex $$n$$ and $$m$$.

Reference: Merzbacher 9.64

EXAMPLES:

sage: x, y = var('x, y')
sage: spherical_harmonic(3, 2, x, y)
15/4*sqrt(7/30)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
15/4*sqrt(7/30)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
sage: spherical_harmonic(3 + I, 2., 1, 2)
-0.351154337307488 - 0.415562233975369*I
sage: latex(spherical_harmonic(3, 2, x, y, hold=True))
Y_{3}^{2}\left(x, y\right)
sage: spherical_harmonic(1, 2, x, y)
0

sage.functions.special.airy_ai(x)

The function $$Ai(x)$$ and the related function $$Bi(x)$$, which is also called an Airy function, are solutions to the differential equation

$y'' - xy = 0,$

known as the Airy equation. The initial conditions $$Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}$$, $$Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}$$ define $$Ai(x)$$. The initial conditions $$Bi(0) = 3^{1/2}Ai(0)$$, $$Bi'(0) = -3^{1/2}Ai'(0)$$ define $$Bi(x)$$.

They are named after the British astronomer George Biddell Airy. They belong to the class of “Bessel functions of fractional order”.

EXAMPLES:

sage: airy_ai(1.0)        # last few digits are random
0.135292416312881400
sage: airy_bi(1.0)        # last few digits are random
1.20742359495287099


REFERENCE:

sage.functions.special.airy_bi(x)

The function $$Ai(x)$$ and the related function $$Bi(x)$$, which is also called an Airy function, are solutions to the differential equation

$y'' - xy = 0,$

known as the Airy equation. The initial conditions $$Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}$$, $$Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}$$ define $$Ai(x)$$. The initial conditions $$Bi(0) = 3^{1/2}Ai(0)$$, $$Bi'(0) = -3^{1/2}Ai'(0)$$ define $$Bi(x)$$.

They are named after the British astronomer George Biddell Airy. They belong to the class of “Bessel functions of fractional order”.

EXAMPLES:

sage: airy_ai(1)        # last few digits are random
0.135292416312881400
sage: airy_bi(1)        # last few digits are random
1.20742359495287099


REFERENCE:

sage.functions.special.elliptic_j(z)

Returns the elliptic modular $$j$$-function evaluated at $$z$$.

INPUT:

• z (complex) – a complex number with positive imaginary part.

OUTPUT:

(complex) The value of $$j(z)$$.

ALGORITHM:

Calls the pari function ellj().

AUTHOR:

John Cremona

EXAMPLES:

   sage: elliptic_j(CC(i))
1728.00000000000
sage: elliptic_j(sqrt(-2.0))
8000.00000000000
sage: z = ComplexField(100)(1,sqrt(11))/2
sage: elliptic_j(z)
-32768.000...
sage: elliptic_j(z).real().round()
-32768

::

sage: tau = (1 + sqrt(-163))/2
sage: (-elliptic_j(tau.n(100)).real().round())^(1/3)
640320

sage.functions.special.error_fcn(t)

The complementary error function $$\frac{2}{\sqrt{\pi}}\int_t^\infty e^{-x^2} dx$$ (t belongs to RR). This function is currently always evaluated immediately.

EXAMPLES:

sage: error_fcn(6)
2.15197367124989e-17
sage: error_fcn(RealField(100)(1/2))
0.47950012218695346231725334611


Note this is literally equal to $$1 - erf(t)$$:

sage: 1 - error_fcn(0.5)
0.520499877813047
sage: erf(0.5)
0.520499877813047

sage.functions.special.hypergeometric_U(alpha, beta, x, algorithm='pari', prec=53)

Default is a wrap of PARI’s hyperu(alpha,beta,x) function. Optionally, algorithm = “scipy” can be used.

The confluent hypergeometric function $$y = U(a,b,x)$$ is defined to be the solution to Kummer’s differential equation

$xy'' + (b-x)y' - ay = 0.$

This satisfies $$U(a,b,x) \sim x^{-a}$$, as $$x\rightarrow \infty$$, and is sometimes denoted x^{-a}2_F_0(a,1+a-b,-1/x). This is not the same as Kummer’s $$M$$-hypergeometric function, denoted sometimes as _1F_1(alpha,beta,x), though it satisfies the same DE that $$U$$ does.

Warning

In the literature, both are called “Kummer confluent hypergeometric” functions.

EXAMPLES:

sage: hypergeometric_U(1,1,1,"scipy")
0.596347362323...
sage: hypergeometric_U(1,1,1)
0.59634736232319...
sage: hypergeometric_U(1,1,1,"pari",70)
0.59634736232319407434...

sage.functions.special.maxima_function(name)

Returns a function which is evaluated both symbolically and numerically via Maxima. In particular, it returns an instance of MaximaFunction.

Note

This function is cached so that duplicate copies of the same function are not created.

EXAMPLES:

sage: spherical_hankel2(2,i)
-e

sage.functions.special.meval(x)

Returns x evaluated in Maxima, then returned to Sage. This is used to evaluate several of these special functions.

TEST:

sage: from sage.functions.special import airy_ai
sage: airy_bi(1.0)
1.207423594952871

sage.functions.special.spherical_bessel_J(n, var, algorithm='maxima')

Returns the spherical Bessel function of the first kind for integers n >= 1.

Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439.

EXAMPLES:

sage: spherical_bessel_J(2,x)
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
sage: spherical_bessel_J(1, 5.2, algorithm='scipy')
-0.12277149950007...
sage: spherical_bessel_J(1, 3, algorithm='scipy')
0.345677499762355...

sage.functions.special.spherical_bessel_Y(n, var, algorithm='maxima')

Returns the spherical Bessel function of the second kind for integers n -1.

Reference: AS 10.1.9 page 437 and AS 10.1.15 page 439.

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: spherical_bessel_Y(2,x)
-((3/x^2 - 1)*cos(x) + 3*sin(x)/x)/x

sage.functions.special.spherical_hankel1(n, var)

Returns the spherical Hankel function of the first kind for integers $$n > -1$$, written as a string. Reference: AS 10.1.36 page 439.

EXAMPLES:

sage: spherical_hankel1(2, x)
(I*x^2 - 3*x - 3*I)*e^(I*x)/x^3

sage.functions.special.spherical_hankel2(n, x)

Returns the spherical Hankel function of the second kind for integers $$n > -1$$, written as a string. Reference: AS 10.1.17 page 439.

EXAMPLES:

sage: spherical_hankel2(2, x)
(-I*x^2 - 3*x + 3*I)*e^(-I*x)/x^3


Here I = sqrt(-1).

Other functions

#### Next topic

Hypergeometric Functions