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Functions that are solutions of the Legendre equation
 
Functions that are solutions of the Legendre equation
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\begin{equation}
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\label{eq1}
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\bigl( 1 - x^2 \bigr)
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\frac{\mathrm{d}^2y}{\mathrm{d}x^2} -
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2x \frac{\mathrm{d}y}{\mathrm{d}x} +
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\left(
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\nu(\nu+1) - \frac{\mu^2}{1-x^2}
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\right)y = 0,
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\end{equation}
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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 $-\nu \leq \mu \leq \nu$, the solutions of \ref{eq1}, restricted to $[-1,1]$, are called Legendre associated functions.
  
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<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>
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====References====  
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{|
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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.
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|-
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|valign="top"|{{Ref|AbSt}}||valign="top"| M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions", Dover, reprint (1965) pp. Chapt. 8
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|-
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|valign="top"|{{Ref|Le}}||valign="top"| N.N. Lebedev, "Special functions and their applications", Dover, reprint (1972) (Translated from Russian)
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====Comments====
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|-
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|}
  
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====References====
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[[Category:Special functions]]
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<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>
 

Latest revision as of 20:14, 1 November 2014


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 $-\nu \leq \mu \leq \nu$, the solutions of \ref{eq1}, restricted to $[-1,1]$, are called Legendre associated functions.

References

[AbSt] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions", Dover, reprint (1965) pp. Chapt. 8
[Le] 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://encyclopediaofmath.org/index.php?title=Legendre_functions&oldid=15230
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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