Sturm-Liouville Theory : Expansion of a Function on an Interval with respect to Eigenfunctions
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Find the expansion of f(x)=1-x on the interval [0,L], with respect to the eigenfunctions of the Sturm-Liouville problem:
y''+λy=0
y(0)=y(L)=0
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Solution Summary
The expansion of function on interval with respect to eigenfunctions.
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The equation has general solution
The boundary condition y(0)=0 implies A=-B, and y(L)=0 implies:
Therefore the solution is non-trivial only if A is not 0 and the difference is 0, that is, only for the eigenvalues
(note ...
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