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'Hermite equation'
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| Title of object: |
Hermite equation |
| Canonical Name: |
HermiteEquation |
| Type: |
Definition |
| Created on: |
2005-05-15 11:28:31 |
| Modified on: |
2005-05-15 17:34:51 |
| Classification: |
msc:34M05 |
| Defines: |
Hermite polynomials |
| Synonyms: |
Hermite equation=Hermite differential equation |
Preamble:
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Content:
The linear differential equation
$$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$
in which $n$ is a real parameter, is called the {\em Hermite equation}. \,The general solution of it is \,$f = Af_1+Bf_2$\, where $f_1$ and $f_2$ have the power series \PMlinkescapetext{presentations}
$$f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+
\frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+...,$$
$$f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+
\frac{2^3(-n)(2-n)(4-n)}{6!}z^6+...$$
converging in the whole complex plane.
If the parameter $n$ 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 \PMlinkescapetext{term} is $(2z)^n$ and called the {\em Hermite polynomials}. |
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