# 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.)

• Jacobi elliptic functions can be thought of as generalizations of both ordinary and hyperbolic trig functions. There are twelve Jacobian elliptic functions. Each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another.

n ------------------- d
|                     |
|                     |
|                     |
s ------------------- c

Each of the corners of the rectangle are labeled, by convention, s, c, d and n. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is on the real axis, and n is on the imaginary axis. The twelve Jacobian elliptic functions are then pq(x), where p and q are one of the letters s,c,d,n.

The Jacobian elliptic functions are then the unique doubly-periodic, meromorphic functions satisfying the following three properties:

1. There is a simple zero at the corner p, and a simple pole at the corner q.
2. The step from p to q is equal to half the period of the function pq(x); that is, the function pq(x) is periodic in the direction pq, with the period being twice the distance from p to q. Also, pq(x) is also periodic in the other two directions as well, with a period such that the distance from p to one of the other corners is a quarter period.
3. If the function pq(x) is expanded in terms of x at one of the corners, the leading term in the expansion has a coefficient of 1. In other words, the leading term of the expansion of pq(x) at the corner p is x; the leading term of the expansion at the corner q is 1/x, and the leading term of an expansion at the other two corners is 1.

We can write

$pq(x)=\frac{pr(x)}{qr(x)}$

where $$p$$, $$q$$, and $$r$$ are any of the letters $$s$$, $$c$$, $$d$$, $$n$$, with the understanding that $$ss=cc=dd=nn=1$$.

Let

$u=\int_0^\phi \frac{d\theta} {\sqrt {1-m \sin^2 \theta}}$

Then the Jacobi elliptic function $$sn(u)$$ is given by

${sn}\; u = \sin \phi$

and $$cn(u)$$ is given by

${cn}\; u = \cos \phi$

and

${dn}\; u = \sqrt {1-m\sin^2 \phi}.$

To emphasize the dependence on $$m$$, one can write $$sn(u,m)$$ for example (and similarly for $$cn$$ and $$dn$$). This is the notation used below.

For a given $$k$$ with $$0 < k < 1$$ they therefore are solutions to the following nonlinear ordinary differential equations:

• $$\mathrm{sn}\,(x;k)$$ solves the differential equations

$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1+k^2) y - 2 k^2 y^3 = 0,$

and

$\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 y^2).$
• $$\mathrm{cn}\,(x;k)$$ solves the differential equations

$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1-2k^2) y + 2 k^2 y^3 = 0,$

and $$\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 + k^2 y^2)$$.

• $$\mathrm{dn}\,(x;k)$$ solves the differential equations

$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - (2 - k^2) y + 2 y^3 = 0,$

and $$\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2= y^2 (1 - k^2 - y^2)$$.

If $$K(m)$$ denotes the complete elliptic integral of the first kind (denoted elliptic_kc), the elliptic functions $$sn (x,m)$$ and $$cn (x,m)$$ have real periods $$4K(m)$$, whereas $$dn (x,m)$$ has a period $$2K(m)$$. The limit $$m\rightarrow 0$$ gives $$K(0) = \pi/2$$ and trigonometric functions: $$sn(x, 0) = \sin x$$, $$cn(x, 0) = \cos x$$, $$dn(x, 0) = 1$$. The limit $$m \rightarrow 1$$ gives $$K(1) \rightarrow \infty$$ and hyperbolic functions: $$sn(x, 1) = \tanh x$$, $$cn(x, 1) = \mbox{\rm sech} x$$, $$dn(x, 1) = \mbox{\rm sech} x$$.

• 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.

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)
0.498011394498832

elliptic_e
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
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.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.inverse_jacobi(sym, x, m)

Here sym = “pq”, where p,q in c,d,n,s. This returns the value of the inverse Jacobi function $$pq^{-1}(x,m)$$. There are a total of 12 functions described by this.

EXAMPLES:

sage: jacobi("sn",1/2,1/2)
jacobi_sn(1/2, 1/2)
sage: float(jacobi("sn",1/2,1/2))
0.4707504736556572
sage: float(inverse_jacobi("sn",0.47,1/2))
0.4990982313222196
sage: float(inverse_jacobi("sn",0.4707504,0.5))
0.4999999114665548
sage: P = plot(inverse_jacobi('sn', x, 0.5), 0, 1, plot_points=20)

Now to view this, just type show(P).

sage.functions.special.jacobi(sym, x, m)

Here sym = “pq”, where p,q in c,d,n,s. This returns the value of the Jacobi function pq(x,m), as described in the documentation for Sage’s “special” module. There are a total of 12 functions described by this.

EXAMPLES:

sage: jacobi("sn",1,1)
tanh(1)
sage: jacobi("cd",1,1/2)
jacobi_cd(1, 1/2)
sage: RDF(jacobi("cd",1,1/2))
0.724009721659
sage: RDF(jacobi("cn",1,1/2)); RDF(jacobi("dn",1,1/2)); RDF(jacobi("cn",1,1/2)/jacobi("dn",1,1/2))
0.595976567672
0.823161001632
0.724009721659
sage: jsn = jacobi("sn",x,1)
sage: P = plot(jsn,0,1)

To view this, type P.show().

sage.functions.special.lngamma(t)

See the log_gamma() function for ‘ documentation and examples.

EXAMPLES:

sage: lngamma(RR(6))
doctest:...: DeprecationWarning: The method lngamma() is deprecated. Use log_gamma() instead.
See http://trac.sagemath.org/6992 for details.
4.78749174278205
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)
1.20742359495
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).

sage.functions.special.spherical_harmonic(m, n, x, y)

Returns the spherical Harmonic function of the second kind for integers $$n > -1$$, $$|m|\leq n$$. 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)

Other functions

Bessel Functions