TITLE: Derivation of Fourier Series?
QUESTION [5 upvotes]: Can someone point me to the full derivation of the Fourier Series? I'm having problems understanding how the a's and b's coeffients are worked out.

REPLY [4 votes]: Imagine that $f(x) = \sum_n a_n \cos n x + \sum_n b_n \sin n x$ (for $x \in [0,2\pi]$, say).  The idea for computing the $a_n$s and $b_n$s is that when
you write down integrals of the form $\int_0^{2\pi} \cos m x \cos n x,$
or $\int_0^{2\pi} \sin m x \sin n x$, or $\int_0^{2 \pi} \cos m x \sin n x$,
then the integrals vanish (just compute them!) unless $m = n$ and the functions
coincide; and in the cases when they don't vanish, their values are easily computed.
So taking $f$, and then computing $\int_0^{2\pi} f(x) \cos n x$ or $\int_0^{2 \pi}
f(x) \sin n x$, one exactly reads off $a_n$ or $b_n$ (for the value of $n$ you
chose).  This is where the formulas come from.
The way people normally think about this is as a kind of orthogonal projection:
the functions $\cos n x$ and $\sin n x$ are like orthogonal basis vectors in 
a vector space, and the integral is like an inner product.  So to find the 
coefficient of a given basis vector (i.e. an $a_n$ or a $b_n$) one takes the
inner product against that particular basis vector.