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Difference between revisions of "Legendre functions"

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Functions that are solutions of the Legendre equation
 
Functions that are solutions of the Legendre equation
 +
\begin{equation}
 +
\label{eq1}
 +
\bigl( 1 - x^2 \bigr)
 +
\frac{\mathrm{d}^2y}{\mathrm{d}x^2} -
 +
2x \frac{\mathrm{d}y}{\mathrm{d}x} +
 +
\left(
 +
\nu(\nu+1) - \frac{\mu^2}{1-x^2}
 +
\right)y = 0,
 +
\end{equation}
 +
where $\nu$ and $\mu$ are arbitrary numbers. If $\nu = 0,1,\ldots$, and $\mu=0$, then the solutions of equation \ref{eq1}, restricted to $[-1,1]$, are called [[Legendre polynomials]]; for integers $\mu$ with $-\nu \leq \mu \les \nu$, the solutions of equation \ref{eq1}, restricted to $[-1,1]$, are called Legendre associated functions.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
====References====  
 
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)</TD></TR></table>
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580303.png" /> are arbitrary numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580305.png" />, then the solutions of equation (*), restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580306.png" />, are called [[Legendre polynomials|Legendre polynomials]]; for integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580307.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580308.png" />, the solutions of equation (*), restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058030/l0580309.png" />, are called Legendre associated functions.
 
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint (1965) pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.N. Lebedev,   "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)</TD></TR></table>
 

Revision as of 20:30, 26 April 2012

Functions that are solutions of the Legendre equation \begin{equation} \label{eq1} \bigl( 1 - x^2 \bigr) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2x \frac{\mathrm{d}y}{\mathrm{d}x} + \left( \nu(\nu+1) - \frac{\mu^2}{1-x^2} \right)y = 0, \end{equation} where $\nu$ and $\mu$ are arbitrary numbers. If $\nu = 0,1,\ldots$, and $\mu=0$, then the solutions of equation \ref{eq1}, restricted to $[-1,1]$, are called Legendre polynomials; for integers $\mu$ with $-\nu \leq \mu \les \nu$, the solutions of equation \ref{eq1}, restricted to $[-1,1]$, are called Legendre associated functions.

References

[a1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) pp. Chapt. 8
[a2] N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)
How to Cite This Entry:
Legendre functions. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Legendre_functions&oldid=25544
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article