Question

When i am reading about Fourier Series, It is stated that any equation can be written as sum of harmonically related exponential components. But i did not understand how we can write this..? can any one help me out...?

Answer 1

This means that any function can be written as a superposition of sines and cosines. I don't recall exactly what the mathematicians call this, but it's because the set of sines and cosines, or harmonically related exponential (since sine and cosine can be written as exponentials), form a basis for the set of all functions. Completeness, I think they call it. If you really really want to see a proof for the completeness of the Fourier series, look around on the internet. I believe I asked my PDEs prof once to see one, and he said he didn't go over it in class because it's a huge pain in the retrice But since this is a physics forum, I'm guessing your using it for E&M or something? It comes up all the time. The only thing you need to know is
\[f(x) = (a_0 / 2) \sum_{n}^{\infty}a_n \cos(nx) + b_n \sin(nx)\]
where n is an integer. That basically says that any function f, can be written as a superposition of sines and cosines, weighted by coefficients a_n. They are harmonic since each coefficient of x is the next subsequent integer. ie
\[f(x) = a_1 \cos(x) + a_2 \cos(2x) + a_3 \cos(3x)...\]
And, of course, sine and cosine can be written in terms of exponentials.