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: