A Legendre polynomial Pn(x) is an even or odd function, depending on whether n is even
Chapter 12, Problem 19(choose chapter or problem)
A Legendre polynomial \(P_{n}(x)\) is an even or odd function, depending on whether n is even or odd. Show that if f is an even function on the interval (-1, 1), then (21) and (22) become, respectively,
\(\begin{gathered}
f(x)=\sum_{n=0}^{\infty} c_{2 n} P_{2 n}(x) \\
c_{2 n}=(4 n+1) \int_{0}^{1} f(x) P_{2 n}(x) d x .
\end{gathered}\)
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