\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:: @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 @c end GSL-manual @page @printindex cp @contents @bye