| Version 12 |
Version 11 |
| The linear differential equation |
The linear differential equation |
| $$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$ |
$$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$ |
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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\\
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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\\
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| \quad $f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+ |
\quad $f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+ |
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\frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+\cdots\!,$\\
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\frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+...,$\\
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| \quad $f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+ |
\quad $f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+ |
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\frac{2^3(-n)(2-n)(4-n)}{6!}z^6+\cdots$\\
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\frac{2^3(-n)(2-n)(4-n)}{6!}z^6+...$\\
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| 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} |
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} |
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$$b_\nu = \frac{2(\nu\!-\!2\!-\!n)}{\nu(\nu\!-\!1)}b_{\nu\!-\!2}.$$
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$$b_\nu = \frac{2(\nu-2-n)}{\nu(\nu-1)}b_{\nu-2}.$$
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| Thus we have the \PMlinkname{radii of convergence}{RadiusOfConvergence} |
Thus we have the \PMlinkname{radii of convergence}{RadiusOfConvergence} |
| $$R = \lim_{\nu\to\infty}\left|\frac{b_{\nu-2}}{b_\nu}\right| = |
$$R = \lim_{\nu\to\infty}\left|\frac{b_{\nu-2}}{b_\nu}\right| = |
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\lim_{\nu\to\infty}\frac{\nu}{2}\!\cdot\!\frac{1\!-\!1/\nu}{1\!-\!(n\!+\!2)/\nu} = \infty.$$
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\lim_{\nu\to\infty}\frac{\nu}{2}\cdot\frac{1-1/\nu}{1-(n+2)/\nu} = \infty.$$
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| Therefore the series converge in the whole complex plane and define entire functions. |
Therefore the series converge in the whole complex plane and define entire functions. |
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| If the \PMlinkescapetext{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 Hermite polynomials. |
If the \PMlinkescapetext{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 Hermite polynomials. |
