Hi Rhys and all,
following up on the open issues for the Hermite polynomials and
functions, please find attached my draft for the documentation. The
introduction is a little longer than in most other chapters, this is on
purpose since I figured that it is easier for one of you to throw parts
of it out than to extend it. So feel free to shorten it, the only piece
of information I find noteworthy is the fact that the Hermite functions
are eigenfunctions of the Fourier transform. This makes them very useful
and is often forgotten.
Since I'm not very well versed in Texinfo, I couldn't get the greek
character psi on the index, so if somebody knows how to fix this, please
do, the corresponding line is commented right now.
Also, the attached file can be compiled on its own, but I've added
comments where the part relevant for the GSL-manual begins and ends.
Cheers,
Konrad
> 3) We'll need to add documentation for the new functions. These will
> resemble content like doc/specfunc-airy.texi. Much of the necessary
> content appears in your email and can be written once we've got a
> gsl_sf_hermite.h in hand from item 1.
\input texinfo @c -*-texinfo-*-
@c %**start of header
@setfilename gsl_sf_hermite.info
@settitle Hermite Polynomials and Functions
@c %**end of header
@setchapternewpage odd
@node Hermite Polynomials and Functions
@chapter Hermite Polynomials and Functions
@tex
\gdef\beforedisplay{}
\gdef\afterdisplay{}
@end tex
@c begin GSL-manual
@cindex Hermite polynomials
@cindex Hermite functions
@cindex @math{He_n(x)}
@cindex @math{H_n(x)}
@c how can you get greek characters in the index in Texinfo?!?
@c @cindex @math{psi_n(x)}
The Hermite polynomials exist in two variants: the probabilists' version @math{He_n(x)} and the physicists'version @math{H_n(x)}. The are defined by the derivatives
@tex
\beforedisplay
$$
\eqalign{
He_n(x) & = (-1)^n e^{x^2/2} \left({d \over dx}\right)^n e^{-x^2/2} \,,\cr
H_n(x) & = (-1)^n e^{x^2} \left({d \over dx}\right)^n e^{-x^2} \,.
}
$$
\afterdisplay
@end tex
@ifinfo
@example
He_n(x) = (-1)^n e^{x^2/2} (d / dx)^n e^{-x^2/2}
H_n(x) = (-1)^n e^{x^2} (d / dx)^n e^{-x^2}
@end example
@end ifinfo
@noindent
They are connected via
@tex
\beforedisplay
$$
\eqalign{
He_n(x) & = 2^{-n/2} H_n({x \over \sqrt{2}}) \,,\cr
H_n(x) & = 2^{n/2} He_n(\sqrt{2} \, x) \,,
}
$$
\afterdisplay
@end tex
@ifinfo
@example
He_n(x) = 2^{-n/2} H_n(x / \sqrt{2})
H_n(x) = 2^{n/2} He_n(\sqrt{2} x)
@end example
@end ifinfo
@noindent
and satisfy the ordinary differential equations
@tex
\beforedisplay
$$
\eqalign{
He_n^{\prime\prime}(x) - x He_n^{\prime}(x) + n He_n(x) & = 0 \,,\cr
H_n^{\prime\prime}(x) - 2x H_n^{\prime}(x) + 2n H_n(x) & = 0 \,.
}
$$
\afterdisplay
@end tex
@ifinfo
@example
He_n^{''}(x) - x He_n^{'}(x) + n He_n(x) = 0
H_n^{''}(x) - 2x H_n^{'}(x) + 2n H_n(x) = 0
@end example
@end ifinfo
@noindent
The closely related Hermite functions are defined by
@tex
\beforedisplay
$$
\psi_n(x) = (n! \sqrt{\pi})^{-1/2} e^{-x^2/2} He_n({\sqrt{2} \, x}) \,,
$$
\afterdisplay
@end tex
@ifinfo
@example
@psi_n = (n! @sqrt{\pi})^{-1/2} e^{-x^2/2} He_n({@sqrt{2} \, x})
@end example
@end ifinfo
@noindent
and satisfy the Schr@"odinger equation for a quantum mechanical harmonic oscillator
@tex
\beforedisplay
$$
\psi_n^{\prime\prime}(x) - (2n + 1 - x^2) \psi_n(x) = 0 \,.
$$
\afterdisplay
@end tex
@ifinfo
@example
@psi_n^{''}(x) - (2n + 1 - x^2) @psi_n(x) = 0
@end example
@end ifinfo
@noindent
Maybe most importantly, the Hermite functions @math{@psi_n} are eigenfunctions of the (continuous) Fourier transform.
For further information see Abramowitz & Stegun, Chapter 22 and Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical Society. The Hermite polynomials and functions are defined in the header file @file{gsl_sf_hermite.h}.
@menu
* Hermite Polynomials::
* Hermite Functions::
* Derivatives of Hermite Polynomials::
* Derivatives of Hermite Functions::
@end menu
@node Hermite Polynomials
@subsection Hermite Polynomials
@deftypefun double gsl_sf_hermite_prob (const int @var{n}, const double @var{x})
This routine evaluates the probabilists' Hermite polynomial @math{He_n(x)} of order n at position x.
@end deftypefun
@deftypefun int gsl_sf_hermite_prob_array (const int @var{nmax}, const double @var{x}, double * @var{result_array})
This routine evaluates all probabilists' Hermite polynomials @math{He_n(x)} up to order nmax at position x. The results are stored in result_array.
@end deftypefun
@deftypefun double gsl_sf_hermite_prob_series (const int @var{n}, const double @var{x}, const double * @var{a})
This routine evaluates the series @math{@sum_{j=0}^n a_j He_j(x)} with @math{He_j} being the j-th probabilists' Hermite polynomial.
@end deftypefun
@deftypefun double gsl_sf_hermite_phys (const int @var{n}, const double @var{x})
This routine evaluates the physicists' Hermite polynomial @math{H_n(x)} of order n at position x.
@end deftypefun
@deftypefun int gsl_sf_hermite_phys_array (const int @var{nmax}, const double @var{x}, double * @var{result_array})
This routine evaluates all physicists' Hermite polynomials @math{H_n} up to order nmax at position x. The results are stored in result_array.
@end deftypefun
@deftypefun double gsl_sf_hermite_phys_series (const int @var{n}, const double @var{x}, const double * @var{a})
This routine evaluates the series @math{@sum_{j=0}^n a_j H_j(x)} with @math{H_j} being the j-th physicists' Hermite polynomial.
@end deftypefun
@node Hermite Functions
@subsection Hermite Functions
@deftypefun double gsl_sf_hermite_func (const int @var{n}, const double @var{x})
This routine evaluates the Hermite function @math{@psi_n(x)} of order n at position x.
@end deftypefun
@deftypefun int gsl_sf_hermite_func_array (const int @var{nmax}, const double @var{x}, double * @var{result_array})
This routine evaluates all Hermite functions @math{@psi_n(x)} up to order nmax at position x. The results are stored in result_array.
@end deftypefun
@deftypefun double gsl_sf_hermite_func_series (const int @var{n}, const double @var{x}, const double * @var{a})
This routine evaluates the series @math{@sum_{j=0}^n a_j @psi_j(x)} with @math{@psi_j} being the j-th Hermite function.
@end deftypefun
@node Derivatives of Hermite Polynomials
@subsection Derivatives of Hermite Polynomials
@deftypefun double gsl_sf_hermite_prob_der (const int @var{m}, const int @var{n}, const double @var{x})
This routine evaluates the m-th derivative of the probabilists' Hermite polynomial @math{He_n(x)} of order n at position x.
@end deftypefun
@deftypefun int gsl_sf_hermite_prob_array_der (const int @var{m}, const int @var{nmax}, const double @var{x}, double * @var{result_array})
This routine evaluates the m-th derivative of all probabilists' Hermite polynomials @math{He_n(x)} up to order nmax at position x. The results are stored in result_array.
@end deftypefun
@deftypefun int gsl_sf_hermite_prob_der_array (const int @var{mmax}, const int @var{n}, const double @var{x}, double * @var{result_array})
This routine evaluates all derivatives (starting from 0) up to the mmax-th derivative of the probabilists' Hermite polynomial of order n @math{He_n(x)} at position x. The results are stored in result_array.
@end deftypefun
@deftypefun double gsl_sf_hermite_phys_der (const int @var{m}, const int @var{n}, const double @var{x})
This routine evaluates the m-th derivative of the physicists' Hermite polynomial @math{H_n(x)} of order n at position x.
@end deftypefun
@deftypefun int gsl_sf_hermite_phys_array_der (const int @var{m}, const int @var{nmax}, const double @var{x}, double * @var{result_array})
This routine evaluates the m-th derivative of all physicists' Hermite polynomials @math{H_n} up to order nmax at position x. The results are stored in result_array.
@end deftypefun
@deftypefun int gsl_sf_hermite_phys_der_array (const int @var{mmax}, const int @var{n}, const double @var{x}, double * @var{result_array})
This routine evaluates all derivatives (starting from 0) up to the mmax-th derivative of the physicists' Hermite polynomial of order n @math{H_n} at position x. The results are stored in result_array.
@end deftypefun
@node Derivatives of Hermite Functions
@subsection Derivatives of Hermite Functions
@deftypefun double gsl_sf_hermite_func_der (const int @var{m}, const int @var{n}, const double @var{x})
This routine evaluates the m-th derivative of the Hermite function @math{@psi_n(x)} of order n at position x.
@end deftypefun
@c end GSL-manual
@page
@printindex cp
@contents
@bye
