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Physical Mathematics by Kevin Cahill

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8

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Legendre functions

8.1 The Legendre polynomials

The monomials xn span the space of functions f(x) that have power-series expansions on an interval about the origin

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(8.1)

They are complete but not orthogonal or normalized. We can make them into real, orthogonal polynomials Pn(x) of degree n on the interval [−1, 1]

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(8.2)

by requiring that each Pn(x) be orthogonal to all monomials xm for m < n

(8.3)

If we impose the ...

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