When a function f(x) can be expressed as a linear combination of a orthogonal basis \phi_n(x) , i.e.

\displaystyle f(x) = \sum_n a_n \phi_n(x)

\displaystyle \langle \phi_n|\phi_m \rangle = \delta_{nm}

then, the integration

\displaystyle \int |f(x)|^2 dx = \sum_n |a_n|^2 \langle \phi_n|\phi_n \rangle = \sum_n |a_n|^2

That is.

Using this theorem, many complicated integration can be calculated as a sum.

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