Miscellaneous special functions

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.

• 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 (cf. Func_assoc_legendre_P), 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:

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

  The spherical harmonics obey the normalisation condition

  \[\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi} Y_\ell^mY_{\ell'}^{m'*}\,d\Omega = \delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega = \sin\theta\,d\varphi\,d\theta .\]

• 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 \(\phi =\pi/2\).

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.

REFERENCES:

• Abramowitz and Stegun: Handbook of Mathematical Functions [AS1964]

• Wikipedia article Spherical_harmonics

• Wikipedia article Helmholtz_equation

• Online Encyclopedia of 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 (2008-02-16): optional calls to scipy and replace all #random by ...

• David Joyner (2008-04-23): addition of elliptic integrals

• Eviatar Bach (2013): making elliptic integrals symbolic

• Eric Gourgoulhon (2022): add Condon-Shortley phase to spherical harmonics

class sage.functions.special.EllipticE

Bases: sage.symbolic.function.BuiltinFunction

Return the incomplete elliptic integral of the second kind:

\[E(\varphi\,|\,m)=\int_0^\varphi \sqrt{1 - m\sin(x)^2}\, dx.\]

EXAMPLES:

sage: z = var("z")
sage: elliptic_e(z, 1)
elliptic_e(z, 1)
sage: # this is still wrong: must be abs(sin(z)) + 2*round(z/pi)
sage: elliptic_e(z, 1).simplify()
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
sage: elliptic_e(1/2, 1/10).n(200)
0.4980113944988315331154610406...

See also

• Taking \(\varphi = \pi/2\) gives elliptic_ec().

• Taking \(\varphi = \operatorname{arc\,sin}(\operatorname{sn}(u,m))\) gives elliptic_eu().

REFERENCES:

• Wikipedia article Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind

• Wikipedia article Jacobi_elliptic_functions

class sage.functions.special.EllipticEC

Bases: sage.symbolic.function.BuiltinFunction

Return the complete elliptic integral of the second kind:

\[E(m)=\int_0^{\pi/2} \sqrt{1 - m\sin(x)^2}\, dx.\]

EXAMPLES:

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

See also

• elliptic_e().

REFERENCES:

• Wikipedia article Elliptic_integral#Complete_elliptic_integral_of_the_second_kind

class sage.functions.special.EllipticEU

Bases: sage.symbolic.function.BuiltinFunction

Return Jacobi’s form of the incomplete elliptic integral of the second kind:

\[E(u,m)= \int_0^u \mathrm{dn}(x,m)^2\, dx = \int_0^\tau \frac{\sqrt{1-m x^2}}{\sqrt{1-x^2}}\, dx.\]

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

Also, elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m).

EXAMPLES:

sage: elliptic_eu (0.5, 0.1)
0.496054551286597

See also

• elliptic_e().

REFERENCES:

• Wikipedia article Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind

• Wikipedia article Jacobi_elliptic_functions

class sage.functions.special.EllipticF

Bases: sage.symbolic.function.BuiltinFunction

Return the incomplete elliptic integral of the first kind.

\[F(\varphi\,|\,m)=\int_0^\varphi \frac{dx}{\sqrt{1 - m\sin(x)^2}},\]

Taking \(\varphi = \pi/2\) gives elliptic_kc().

EXAMPLES:

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

See also

• elliptic_e().

REFERENCES:

• Wikipedia article Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind

class sage.functions.special.EllipticKC

Bases: sage.symbolic.function.BuiltinFunction

Return the complete elliptic integral of the first kind:

\[K(m)=\int_0^{\pi/2} \frac{dx}{\sqrt{1 - m\sin(x)^2}}.\]

EXAMPLES:

sage: elliptic_kc(0.5)
1.85407467730137

See also

• elliptic_f().

• elliptic_ec().

REFERENCES:

• Wikipedia article Elliptic_integral#Complete_elliptic_integral_of_the_first_kind

• Wikipedia article Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind

class sage.functions.special.EllipticPi

Bases: sage.symbolic.function.BuiltinFunction

Return the incomplete elliptic integral of the third kind:

\[\Pi(n, t, m) = \int_0^t \frac{dx}{(1 - n \sin(x)^2)\sqrt{1 - m \sin(x)^2}}.\]

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)

REFERENCES:

• Wikipedia article Elliptic_integral#Incomplete_elliptic_integral_of_the_third_kind

class sage.functions.special.SphericalHarmonic

Bases: sage.symbolic.function.BuiltinFunction

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 \(m\).

EXAMPLES:

sage: x, y = var('x, y')
sage: spherical_harmonic(3, 2, x, y)
1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
1/8*sqrt(30)*sqrt(7)*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

The degree \(n\) and the order \(m\) can be symbolic:

sage: n, m = var('n m')
sage: spherical_harmonic(n, m, x, y)
spherical_harmonic(n, m, x, y)
sage: latex(spherical_harmonic(n, m, x, y))
Y_{n}^{m}\left(x, y\right)
sage: diff(spherical_harmonic(n, m, x, y), x)
m*cot(x)*spherical_harmonic(n, m, x, y)
 + sqrt(-(m + n + 1)*(m - n))*e^(-I*y)*spherical_harmonic(n, m + 1, x, y)
sage: diff(spherical_harmonic(n, m, x, y), y)
I*m*spherical_harmonic(n, m, x, y)

The convention regarding the Condon-Shortley phase \((-1)^m\) is the same as for SymPy’s spherical harmonics and Wikipedia article Spherical_harmonics:

sage: spherical_harmonic(1, 1, x, y)
-1/4*sqrt(3)*sqrt(2)*e^(I*y)*sin(x)/sqrt(pi)
sage: from sympy import Ynm
sage: Ynm(1, 1, x, y).expand(func=True)
-sqrt(6)*exp(I*y)*sin(x)/(4*sqrt(pi))
sage: spherical_harmonic(1, 1, x, y) - Ynm(1, 1, x, y)
0

It also agrees with SciPy’s spherical harmonics:

sage: spherical_harmonic(1, 1, pi/2, pi).n()  # abs tol 1e-14
0.345494149471335
sage: from scipy.special import sph_harm  # NB: arguments x and y are swapped
sage: sph_harm(1, 1, pi.n(), (pi/2).n())  # abs tol 1e-14
(0.3454941494713355-4.231083042742082e-17j)

Note that this convention differs from the one in Maxima, as revealed by the sign difference for odd values of \(m\):

sage: maxima.spherical_harmonic(1, 1, x, y).sage()
1/2*sqrt(3/2)*e^(I*y)*sin(x)/sqrt(pi)

It follows that, contrary to Maxima, SageMath uses the same sign convention for spherical harmonics as SymPy, SciPy, Mathematica and Wikipedia article Table_of_spherical_harmonics.

REFERENCES:

• Wikipedia article Spherical_harmonics

sage.functions.special.elliptic_eu_f(u, m)

Internal function for numeric evaluation of elliptic_eu, defined as \(E\left(\operatorname{am}(u, m)|m\right)\), where \(E\) is the incomplete elliptic integral of the second kind and \(\operatorname{am}\) is the Jacobi amplitude function.

EXAMPLES:

sage: from sage.functions.special import elliptic_eu_f
sage: elliptic_eu_f(0.5, 0.1)
mpf('0.49605455128659691')

sage.functions.special.elliptic_j(z, prec=53)

Returns the elliptic modular \(j\)-function evaluated at \(z\).

INPUT:

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

• prec (default: 53) – precision in bits for the complex field.

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

This example shows the need for higher precision than the default one of the \(ComplexField\), see trac ticket #28355:

sage: -elliptic_j(tau) # rel tol 1e-2
2.62537412640767e17 - 732.558854258998*I
sage: -elliptic_j(tau,75) # rel tol 1e-2
2.625374126407680000000e17 - 0.0001309913593909879441262*I
sage: -elliptic_j(tau,100) # rel tol 1e-2
2.6253741264076799999999999999e17 - 1.3012822400356887122945119790e-12*I
sage: (-elliptic_j(tau, 100).real().round())^(1/3)
640320