Difference between revisions of "Legendre functions"
From Encyclopedia of Mathematics
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Functions that are solutions of the Legendre equation | Functions that are solutions of the Legendre 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 | + | 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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| + | |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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| + | |valign="top"|{{Ref|Le}}||valign="top"| N.N. Lebedev, "Special functions and their applications", Dover, reprint (1972) (Translated from Russian) | ||
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| + | [[Category:Special functions]] | ||
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=25545
Legendre functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Legendre_functions&oldid=25545
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article