Talk:PlanetPhysics/Hermite Equation

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Hermite equation %%% Primary Category Code: 02.30.Hq %%% Filename: HermiteEquation.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

The linear \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} $$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$

in which $n$ is a real constant, is called the {\em Hermite equation}.\, Its general solution is\, $f := Af_1\!+\!Bf_2$\, with $A$ and $B$ arbitrary constants and the \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} $f_1$ and $f_2$ presented as\\

\quad $f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+ \frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+\cdots\!,$\\

\quad $f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+ \frac{2^3(-n)(2-n)(4-n)}{6!}z^6+\cdots$\\

It's easy to check that these \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} series satisfy the differential equation.\, The coefficients $b_\nu$ in both series obey the recurrence formula $$b_\nu = \frac{2(\nu\!-\!2\!-\!n)}{\nu(nu\!-\!1)}b_{\nu\!-\!2}.$$ Thus we have the radii of convergence $$R = \lim_{\nu\to\infty}\left|\frac{b_{\nu-2}}{b_\nu}\right| = \lim_{\nu\to\infty}\frac{\nu}{2}\!\cdot\!\frac{1\!-\!1/\nu}{1\!-\!(n\!+\!2)/\nu} = \infty.$$ Therefore the series converge in the whole complex plane and define entire functions.

If the constant $n$ is a non-negative integer, then one of $f_1$ and $f_2$ is simply a polynomial function.\, The polynomial solutions of the Hermite equation are usually normed so that the highest degree term is $(2z)^n$ and called the \htmladdnormallink{Hermite polynomials}{http://planetphysics.us/encyclopedia/HermitePolynomials.html}.

\end{document}