Eigenfunction Expansions
Eigenfunction Expansions
In this special case we are considering, p(x)=1 and we have two sets of eigenfunctions, un and vn. We will use a special eigenfunction expansion, called the FOURIER EXPANSION or FOURIER SERIES, that is a combination of the two eigenfunction expansions.
Make note of a few things. The eigenfunctions are un=Cos[nPi x/L] and vn=Sin[nPi x/L]. The eigenfunction u0=1 is used but v0=0 is not. We have already noticed that the integrals in the denominators satisfy
With this in mind, one often sees the Fourier series written as
One of the ideas behind Fourier series is to obtain every
periodic function (even discontinuous ones) of period 2L as a combination of
un=Cos[nPi x/L] and vn=Sin[nPi x/L], in a sense the "pure" periodic
functions, sine and cosine.
If a function is not periodic, we must redefine it so that it is (see the
sections below on periodic extensions).