Difference between revisions of "Legendre functions"
From Encyclopedia of Mathematics
(TeXed, Refs, tidied) |
|||
| Line 12: | Line 12: | ||
\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 $-\nu \leq \mu \leq \nu$, the solutions of | + | 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==== | ====References==== | ||
Revision as of 19:48, 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 \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=25549
Legendre functions. Encyclopedia of Mathematics. URL: http://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