Apologies for the textual format (no idea how to write these equations properly in a forum) but I hope those who are familiar with Fourier Series will recognise these equations.

Q: In Fourier Series, our teacher has told us that it should be obvious to us whether to use the synthesis :

x(t) = inf.sum of ak*e^(jkw0t)

or the analysis equation :

ak = 1/T0 * integral of x(t)*e^(-jkw0t) dt

But I'm afraid I haven't got the foggiest myself when to use one and not the other and it is not explained anywhere in the notes we have been given or in numerous textbooks I have checked.

Wiki suggests something about one is to break up terms and the other is the restructure but that is not the context my teacher in the engineering course is putting them in and I suspect it is wiki-garbage.

Please can someone help before I start banging my head against the wall with intent of permanent coma.