Validity of Fourier Series Expansion for Non-Periodic Functions

[sandylam966]

Homework Statement

Given ∑$^{∞}_{n=1}$ n A_n sin($\frac{n\pi x}{L}$) = $\frac{λL}{\pi c}$ σ(x-$\frac{L}{2}$) + A sin($\frac{\pi x}{2}$), where L, λ, c, σ and A are known constants, find A_n.

Homework Equations

Fourier half-range sine expansion.

The Attempt at a Solution

I understand I should expand the RHS as an odd function with period (-L, L) and then compare the coefficients with the LHS, and I do get to correct result. However I didn't understand why I could do so. I mean, originally RHS is NOT a periodic function, that it certainly does not equal the 'constructed' Fourier sine expansion. So how could the coefficients equal since it's actually a different function?

 

[HallsofIvy]

How do you know you get the "correct result"? Do you mean you get the result given in your text?

If so then the given answer is NOT for the given function but for a function defined to be $\frac{\lambda L}{\pi c}\sigma(x-\frac{L}{2})+ A sin(\frac{\pi x}{2})$ on (-L, L) and continued "by periodicity" to the rest of the real numbers.