# Eigenfunction expansion solution to a PDE with a constant non homogeneous term

I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogeneous term is a constant, rather than a function of the independent variables? For example, in a hyperbolic PDE with x and t as the independent variables the eigenfunctions might be something like $\sum_{n=1}^\infty sin(n\pi x)$, and to create an eigenfunction expansion of the nonhomogeneous term I have to solve for the coefficients A of $\sum_{n=1}^\infty A sin(n\pi x) = B$ by using the orthogonality of sines, where B is the constant nonhomogeneous term. I guess I'm having trouble seeing how an infinite series of sines could converge to a constant - I know in a Fourier series one has to solve for the "DC component" separately.

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## 1 Answer

An infinite series of sines can indeed converge to a constant, in the interior of the interval in question (I guess you have Dirichlet boundary conditions at the endpoints). If you look at the sum of that Fourier series on the whole real line you will see a square wave, where the part that you're interested in constitutes half a period.

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I see now - so long as the series converges to the constant within the boundary conditions of the problem it doesn't matter that it doesn't equal the constant elsewhere. Thanks! –  Bitrex Feb 27 '11 at 21:47
Exactly. You're welcome! :) –  Hans Lundmark Feb 27 '11 at 22:00