Legendre functions

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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.


[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)
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Legendre functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article