\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

@c Version 1: Konrad Griessinger (konradg(at)gmx.net), 12/2013

@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(address@hidden \, 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 @address@hidden 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::  
* Zeros of Hermite Polynomials and Hermite Functions:: 
* Gauss-Hermite quadrature::  
@end menu

@node Hermite Polynomials
@subsection Hermite Polynomials


@deftypefun double gsl_sf_hermite_prob (const int @var{n}, const double @var{x})
@deftypefunx int gsl_sf_hermite_prob_e (const int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate 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})
@deftypefunx int gsl_sf_hermite_prob_series_e (const int @var{n}, const double @var{x}, const double * @var{a}, gsl_sf_result * @var{result})
These routines evaluate the series @address@hidden a_j He_j(x)} with @math{He_j} being the j-th probabilists' Hermite polynomial using the Clenshaw algorithm.
@end deftypefun

@deftypefun double gsl_sf_hermite_phys (const int @var{n}, const double @var{x})
@deftypefunx int gsl_sf_hermite_phys_e (const int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate 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})
@deftypefunx int gsl_sf_hermite_phys_series_e (const int @var{n}, const double @var{x}, const double * @var{a}, gsl_sf_result * @var{result})
These routines evaluate the series @address@hidden a_j H_j(x)} with @math{H_j} being the j-th physicists' Hermite polynomial using the Clenshaw algorithm.
@end deftypefun


@node Hermite Functions
@subsection Hermite Functions


@deftypefun double gsl_sf_hermite_func (const int @var{n}, const double @var{x})
@deftypefunx int gsl_sf_hermite_func_e (const int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate the Hermite function @address@hidden(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 @address@hidden(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})
@deftypefunx int gsl_sf_hermite_func_series_e (const int @var{n}, const double @var{x}, const double * @var{a}, gsl_sf_result * @var{result})
These routines evaluate the series @address@hidden a_j @psi_j(x)} with @address@hidden being the j-th Hermite function using the Clenshaw algorithm.
@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})
@deftypefunx int gsl_sf_hermite_prob_der_e (const int @var{m}, const  int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate 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})
@deftypefunx int gsl_sf_hermite_phys_der_e (const int @var{m}, const int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate 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})
@deftypefunx int gsl_sf_hermite_func_der_e (const int @var{m}, const int @var{n}, const double @var{x}, gsl_sf_result * @var{result})
These routines evaluate the m-th derivative of the Hermite function @address@hidden(x)} of order n at position x.
@end deftypefun


@node Zeros of Hermite Polynomials and Hermite Functions
@subsection Zeros of Hermite Polynomials and Hermite Functions


These routines calculate the @var{s}-th zero of the Hermite Polynomial/Function of order @var{n}. Since the zeros are symmetrical around zero, only positive zeros are calculated, ordered from smallest to largest, starting from index 1. Only for odd polynomial orders a zeroth zero exists, its value always being zero.


@deftypefun double gsl_sf_hermite_prob_zero (const int @var{n}, const int @var{s})
@deftypefunx int gsl_sf_hermite_prob_zero_e (const int @var{n}, const int @var{s}, gsl_sf_result * @var{result})
These routines evaluate the @var{s}-th zero of the probabilists' Hermite polynomial @math{He_n(x)} of order @var{n}.
@end deftypefun


@deftypefun double gsl_sf_hermite_phys_zero (const int @var{n}, const int @var{s})
@deftypefunx int gsl_sf_hermite_phys_zero_e (const int @var{n}, const int @var{s}, gsl_sf_result * @var{result})
These routines evaluate the @var{s}-th zero of the physicists' Hermite polynomial @math{H_n(x)} of order @var{n}.
@end deftypefun


@deftypefun double gsl_sf_hermite_func_zero (const int @var{n}, const int @var{s})
@deftypefunx int gsl_sf_hermite_func_zero_e (const int @var{n}, const int @var{s}, gsl_sf_result * @var{result})
These routines evaluate the @var{s}-th zero of the Hermite function @address@hidden(x)} of order @var{n}.
@end deftypefun


@c end GSL-manual

@page
@printindex cp

@contents
@bye