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.
References
| [1] | M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions", Dover, reprint (1965) pp. Chapt. 8 |
| [2] | N.N. Lebedev, "Special functions and their applications", Dover, reprint (1972) (Translated from Russian) |
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