Origin of Fourier Series
You might wonder why the Fourier Series coefficients have the form they have. Consider the general case of trigonometric series:
Integrating we have:
This factor is the average value of f(t). So if f(t) positioned symmetrically over the t axis (odd function), then the average value is zero.
Multiplying the general trigonometric series by cos(nωt) and sin(nωt) and integrating we have:
And since it is:
Substituting we have:
Changing variable k to n, so that we have only one notation for the variable: