# Legendre functions

From Encyclopedia of Mathematics

Functions that are solutions of the Legendre equation
\begin{equation}
\tag{1}
\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 (1), restricted to $[-1,1]$, are called Legendre polynomials; for integers $\mu$ with $-\nu \leq \mu \leq \nu$, the solutions of (1), 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://www.encyclopediaofmath.org/index.php?title=Legendre_functions&oldid=25549

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