Following Fourier, any periodic function y(t) which repeats itself with a period T:
y(t) = y (t + nT), where n = 0, ±1, ±2, ±3...
can be expanded in a Fourier series:
y(t) = y0 + Sn [Cn cos((n2p/T) . t) + Sn sin((n2p/T) . t)) ]
Where Cn and Sn are the appropriate Fourier coefficients. The term T is the period of the expansion.
Using the Euler equation exp(i t) = cos(t) + i sin(t) the last Fourier expansion can be also written as:
y(t) = Sn [En exp((n2p/T) . t) ]
If angular frequency is used in the expansion, the term (n2p/T) . t) yields just nw . t. instead of n2pf . t, which makes the equations simpler.
Fourier transformation is used to obtain the values of coefficients Cn, Sn or En and therefore reveal the periodicity of a function since the result gives us the amplitudes and the frequencies of the simple harmonic (sin and cos) functions which, when combined, produce the complex periodic function y(t). This process is also called Fourier analysis of a complex function.
Inverse Fourier transformation is used to obtain the complex periodic function y(t) if the frequencies and corresponding amplitudes are already known. It can be called Fourier synthesis of a complex function from simple periodic ones.