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Proposition: Legendre Polynomials and Legendre Differential Equations

The $n$-th order Legendre polynomial is defined by $$P_{n}(x)={1 \over 2^{n}n!}{d^{n} \over dx^{n}}\left[(x^{2}-1)^{n}\right].$$

It is the solution of the following ordinary Legendre differential equations:

$$(1-x^2)P_n^{\prime\prime}(x)-2xP_n^\prime(x)+n(n+1)P_{n}(x)=0.$$

The equations are named after Adrien-Marie Legendre (1752 - 1833).

Table of Contents

Proofs: 1

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983