@menu
* Introduction to Orthogonal Polynomials::
* Definitions for Orthogonal Polynomials::
@end menu
@node Introduction to Orthogonal Polynomials, Definitions for Orthogonal Polynomials, , Orthogonal Polynomials
@comment node-name, next, previous, up
@section Introduction to Orthogonal Polynomials
The specfun package, located in the share directory, contains Maxima
code for the evaluation of all orthogonal polynomials listed in
Chapter 22 of Abramowitz and Stegun. These include Chebyshev,
Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer)
polynomials. Additionally, specfun contains code for
spherical Bessel, spherical Hankel, and spherical harmonic functions.
The following table lists each function in specfun,
its Maxima name, restrictions on its arguments
(@math{ m } and @math{ n } must be integers), and a
reference to the algorithm specfun uses to evaluate it.
With few exceptions, specfun follows the conventions of
Abramowitz and Stegun. Before you use specfun, check
that specfun's conventions match your expectations.
A&S refers to Abramowitz and Stegun, @emph{Handbook of
Mathematical Functions} (10th printing, December 1972),
G&R to Gradshteyn and Ryzhik,
@emph{Table of Integrals, Series, and Products}
(1980 corrected and enlarged edition), and Merzbacher
to @emph{Quantum Mechanics} (2ed, 1970).
@multitable @columnfractions 0.25 0.25 0.25 0.25
@item @emph{Function} @tab @emph{Maxima Name} @tab @emph{Restrictions} @tab @emph{Reference(s)}
@item @ Chebyshev T @tab chebyshev_t(n, x) @tab @math{n > -1} @tab A&S 22.5.31
@item Chebyshev U @tab chebyshev_u(n, x) @tab @math{n > -1} @tab A&S 22.5.32
@item generalized Laguerre @tab gen_laguerre(n,a,x) @tab @math{n > -1} @tab A&S page 789
@item Laguerre @tab laguerre(n,x) @tab @math{n > -1} @tab A&S 22.5.67
@item Hermite @tab hermite(n,x) @tab @math{ n > -1} @tab A&S 22.4.40, 22.5.41
@item Jacobi @tab jacobi_p(n,a,b,x) @tab @math{ n > -1, a, b > -1} @tab A&S page 789
@item associated Legendre P @tab assoc_legendre_p(n,m,x) @tab @math{n > -1} @tab A&S 22.5.37, 8.6.6, 8.2.5
@item associated Legendre Q @tab assoc_legendre_q(n,m,x) @tab @math{n > -1, m > -1} @tab G & R 8.706
@item Legendre P @tab legendre_p(n,m,x) @tab @math{n > -1} @tab A&S 22.5.35
@item Legendre Q @tab legendre_q(n,m,x) @tab @math{n > -1} @tab A&S 8.6.19
@item spherical Hankel 1st @tab spherical_hankel1(n, x) @tab @math{n > -1} @tab A&S 10.1.36
@item spherical Hankel 2nd @tab spherical_hankel2(n, x) @tab @math{n > -1} @tab A&S 10.1.17
@item spherical Bessel J @tab spherical_bessel_j(n,x) @tab @math{n > -1} @tab A&S 10.1.8, 10.1.15
@item spherical Bessel Y @tab spherical_bessel_y(n,x) @tab @math{n > -1} @tab A&S 10.1.9, 10.1.15
@item spherical harmonic @tab spherical_harmonic(n,m,x,y) @tab @math{n > -1, |m| <= n} @tab Merzbacher 9.64
@item ultraspherical (Gegenbauer) @tab ultraspherical(n,a,x) @tab @math{ n > -1} @tab A&S 22.5.27
@end multitable
The specfun package is primarily intended for symbolic
computation. It is hoped that it gives accurate floating point
results as well; however, no claims are made that the algorithms
are well suited for numerical evaluation. Some effort, however,
has been made to provide good numerical performance.
When all arguments, except for the order, are floats (but not bfloats),
many functions in specfun call a float modedeclared version of the
Jacobi function. This greatly speeds floating point evaluation
of the orthogonal polynomials.
specfun handles most domain errors by returning an unevaluated
function. No attempt has been made to define simplification
rules (based on recursion relations) for unevaluated functions.
Users should be aware that it is possible for an expression
involving sums of unevaluated special functions to vanish, yet
Maxima is unable to reduce it to zero. Be careful.
To access functions in specfun, you must first load specfun.o.
Alternatively, you may append autoload statements to your
init.lsp file (located in your working directory). To autoload the
hermite function, for example, append
@lisp
(defprop |$hermite| #"specfun.o" autoload)
(add2lnc '|$hermite| $props)
@end lisp
to your init.lsp file. An example use of specfun is
@example
(c1) load("specfun.o")$
(c2) [hermite(0,x),hermite(1,x),hermite(2,x)];
(d2) [1,2*x,-2*(1-2*x^2)]
(c3) diff(hermite(n,x),x);
(d3) 2*n*hermite(n-1,x)
@end example
When using the compiled version of specfun, be especially
careful to use the correct number of function arguments; calling
them with too few arguments may generate a fatal error messages.
For example
@example
(c1) load("specfun")$
/* chebyshev_t requires two arguments. */
(c2) chebyshev_t(8);
Error: Caught fatal error [memory may be damaged]
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MMAPCAR.
Broken at SIMPLIFY. Type :H for Help.
@end example
Maxima code translated into Lisp handles such errors more gracefully.
If specfun.LISP is installed on your machine, the same computation
results in a clear error message. For example
@example
(c1) load("specfun.LISP")$
(c2) chebyshev_t(8);
Error: Expected 2 args but received 1 args
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MACSYMA-TOP-LEVEL.
Broken at |$CHEBYSHEV_T|. Type :H for Help.
@end example
Generally, compiled code runs faster than translated code; however,
translated code may be better for program development.
For some functions, when the order is symbolic but has been
declared to be an integer, specfun will return a series
representation. (The series representation is not
used by specfun for any computations.) You may use this feature to
find symbolic values for special values orthogonal polynomials.
An example:
@example
(c1) load("specfun")$
(c2) legendre_p(n,1);
(d2) legendre_p(n, 1)
/* Declare n to be an integer; now legendre_p(n,1) evaluates to 1. */
(c3) declare(n,integer)$
(c4) legendre_p(n,1);
(d4) 1
(c5) ultraspherical(n,3/2,1);
(d4) (n+1)*gamma (n+3) / (2*gamma (n+2))
@end example
Although the preceding example doesn't show it, two terms
of the sum are added outside the summation. Removing these
two terms avoids errors associated with @math{ 0^0} terms
in a sum that should evaluate to 1, but evaluate to 0 in a Maxima
summation. Because the sum index runs from 1 to
@math{n - 1}, the lower sum
index will exceed the upper sum index when @math{ n = 0};
setting sumhack to true provides a fix. For example:
@example
(c1) load("specfun.o")$
(c2) declare(n,integer)$
(c3) e : legendre_p(n,x)$
(c4) ev(e,sum,n=0);
Lower bound to SUM: 1
is greater than the upper bound: - 1
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);)
(c5) ev(e,sum,n=0),sumhack : true;
(d5) 1
@end example
Most functions in specfun have a gradef property;
derivatives with respect to the order or other function parameters
aren't unevaluated.
The specfun package and its documentation were written by
Barton Willis of the University of Nebraska at Kearney. It is
released under the terms of the General Public License (GPL).
Send bug reports and comments on this package to
willisb@@unk.edu. In your report, please include
Maxima and specfun version information. The specfun
version may be found using get:
@example
(c2) get('specfun,'version);
(d2) 110
@end example
@c end concepts Orthogonal Polynomials
@node Definitions for Orthogonal Polynomials, , Introduction to Orthogonal Polynomials, Orthogonal Polynomials
@section Definitions for Orthogonal Polynomials
@defun ASSOC_LEGENDRE_P (n, m, x)
[specfun package] return the associated Legendre function
of the first kind for integers @math{ n > -1 } and
@math{ m > -1}. When @math{ | m | > n } and @math{ n >= 0},
we have @math{assoc_legendre_p (n, m, x) = 0}.
Reference: A&S 22.5.37 page 779, A&S 8.6.6 (second
equation) page 334, and A&S 8.2.5 page 333.
To access this function, load("specfun").
See @ref{ASSOC_LEGENDRE_Q}, @ref{LEGENDRE_P}, and @ref{LEGENDRE_Q}.
@end defun
@anchor{ASSOC_LEGENDRE_Q}
@defun ASSOC_LEGENDRE_Q (n, m, x)
[specfun package] return the associated Legendre function
of the second kind for integers @math{ n > -1 } and
@math{ m > -1}.
Reference: Gradshteyn and Ryzhik 8.706 page 1000.
To access this function, load("specfun").
See also ASSOC_LEGENDRE_P, LEGENDRE_P, and LEGENDRE_Q.
@end defun
@defun CHEBYSHEV_T (n, x)
[specfun package] return the Chebyshev function of the first kind
for integers @math{ n > -1}.
Reference: A&S 22.5.31 page 778 and A&S 6.1.22 page 256.
To access this function, load("specfun").
See also CHEBYSHEV_U.
@end defun
@defun CHEBYSHEV_U (n, x)
[specfun package] return the Chebyshev function of the
second kind for integers @math{ n > -1}.
Reference: A&S, 22.8.3 page 783 and A&S 6.1.22 page 256.
To access this function, load("specfun").
See also CHEBYSHEV_T.
@end defun
@defun GEN_LAGUERRE (n, a, x)
[specfun package] return the generalized Laguerre polynomial
for integers @math{ n > -1}.
To access this function, load("specfun").
Reference: table on page 789 in A&S.
@end defun
@defun HERMITE (n,x)
[specfun package] return the Hermite polynomial for
integers @math{ n > -1}.
To access this function, load("specfun").
Reference: A&S 22.5.40 and 22.5.41, page 779.
@end defun
@defun JACOBI_P (n, a, b, x)
[specfun package] return the Jacobi polynomial for
integers @math{ n > -1 } and @math{ a } and @math{ b }
symbolic or @math{ a > -1} and @math{b > -1}.
(The Jacobi polynomials are actually defined for all
@math{ a } and @math{ b }; however, the Jacobi polynomial
weight @math{ (1-x)^a(1+x)^b} isn't integrable for @math{ a <= -1} or
@math{ b <= -1}. )
When @math{a, b}, and @math{ x } are floats (but not bfloats)
specfun calls a special modedeclared version of @math{jacobi_p}.
For numerical values, the modedeclared version is much
faster than the other version. Many functions in specfun are
computed as a special case of the Jacobi polynomials; they also
enjoy the speed boost from the modedeclared version of
@math{ jacobi}.
If @math{ n } has been declared to be an integer,
@math{jacobi_p (n, a, b, x)} returns a summation representation
for the Jacobi function. Because Maxima simplifies
@math{0^0} to @math{ 0 } in a sum, two terms of the sum are
added outside the summation.
To access this function, load("specfun").
Reference: table on page 789 in A&S.
@end defun
@defun LAGUERRE (n, x)
[specfun package] return the Laguerre polynomial for
integers @math{ n > -1}.
Reference: A&S 22.5.16, page 778 and A&S page 789.
To access this function, load("specfun").
See also GEN_LAGUERRE.
@end defun
@anchor{LEGENDRE_P}
@defun LEGENDRE_P (n, x)
[specfun package] return the Legendre polynomial of the
first kind for integers @math{ n > -1}.
Reference: A&S 22.5.35 page 779.
To access this function, load("specfun").
See @ref{LEGENDRE_Q}.
@end defun
@anchor{LEGENDRE_Q}
@defun LEGENDRE_Q (n, x)
[specfun package] return the Legendre polynomial of the
first kind for integers @math{ n > -1}.
Reference: A&S 8.6.19 page 334.
To access this function, load("specfun").
See also LEGENDRE_P.
@end defun
@defun SPHERICAL_BESSEL_J (n, x)
[specfun package] return the spherical Bessel function of
the first kind for integers @math{ n > -1}.
Reference: A&S 10.1.8 page 437 and A&S 10.1.15 page 439.
To access this function, load("specfun").
See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2,
and SPHERICAL_BESSEL_Y.
@end defun
@defun SPHERICAL_BESSEL_Y (n, x)
[specfun package] return the spherical Bessel function of
the second kind for integers @math{ n > -1}.
Reference: A&S 10.1.9 page 437 and 10.1.15 page 439.
To access this function, load("specfun").
See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2,
and SPHERICAL_BESSEL_Y.
@end defun
@defun SPHERICAL_HANKEL1 (n,x)
[specfun package] return the spherical hankel function of the
first kind for integers @math{ n > -1}.
Reference: A&S 10.1.36 page 439.
To access this function, load("specfun").
See also SPHERICAL_HANKEL2, SPHERICAL_BESSEL_J, and
SPHERICAL_BESSEL_Y.
@end defun
@defun SPHERICAL_HANKEL2 (n,x)
[specfun package] return the spherical hankel function of the
second kind for integers @math{ n > -1}.
Reference: A&S 10.1.17 page 439.
To access this function, load("specfun").
See also SPHERICAL_HANKEL1, SPHERICAL_BESSEL_J, and
SPHERICAL_BESSEL_Y.
@end defun
@defun SPHERICAL_HARMONIC (n, m, x, y)
[specfun package] return the spherical harmonic function
for integers @math{ n > -1} and @math{ | m | <= n }.
Reference: Merzbacher 9.64.
To access this function, load("specfun").
See also ASSOC_LEGENDRE_P
@end defun
@defun ULTRASPHERICAL (n,a,x)
[specfun package] return the ultraspherical polynomials for
integers @math{ n > -1}. The ultraspherical polynomials are
also known as Gegenbauer polynomials.
Reference: A&S 22.5.27
To access this function, load("specfun").
See also JACOBI_P.
@end defun