# Difference between revisions of "Legendre functions"

From Encyclopedia of Mathematics

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\right)y = 0, | \right)y = 0, | ||

\end{equation} | \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 \ | + | 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 equation \ref{eq1}, restricted to $[-1,1]$, are called Legendre associated functions. |

====References==== | ====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> | <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 19:30, 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 $-\nu \leq \mu \leq \nu$, 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://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