|
|
|
Viewing Version
8
of
'Hermite equation'
|
[ view 'Hermite equation'
|
back to history
]
| Title of object: |
Hermite equation |
| Canonical Name: |
HermiteEquation |
| Type: |
Definition |
| Created on: |
2005-05-15 11:28:31 |
| Modified on: |
2005-05-17 09:03:13 |
| Classification: |
msc:34M05 |
| Synonyms: |
Hermite equation=Hermite differential equation |
Preamble:
% this is the default PlanetMath 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
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem} |
Content:
The linear differential equation
$$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$
in which $n$ is a real \PMlinkescapetext{constant}, is called the {\em Hermite equation}. \,Its general solution is \,$f := Af_1+Bf_2$\, with $A$ and $B$ arbitrary \PMlinkescapetext{constants} and the functions $f_1$ and $f_2$ presented as\\
\quad $f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+
\frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+...,$\\
\quad $f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+
\frac{2^3(-n)(2-n)(4-n)}{6!}z^6+...$\\
It's easy to check that these power series satisfy the differential equation. \,The coefficients $b_\nu$ in both series obey the recurrence \PMlinkescapetext{formula}
$$b_\nu = \frac{2(\nu-2-n)}{\nu-1}b_{\nu-2}.$$
Thus we have the \PMlinkname{radii of convergence}{RadiusOfConvergence}
$$R = \lim_{\nu\to\infty}|\frac{b_{\nu-2}}{b_\nu}| =
\lim_{\nu\to\infty}\frac{\nu}{2}\cdot\frac{1-1/\nu}{1-(n+2)/\nu} = \infty.$$
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 \PMlinkname{degree}{PolynomialRing} \PMlinkescapetext{term} is $(2z)^n$ and called the {\em Hermite polynomials}. |
|
|
|
|
|
