Talk:PlanetPhysics/Hermite Equation
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%%% 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}
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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}.
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