Exponential Fourier Series

You can write the Fourier series using complex exponential functions. By using Euler's formula we can rewrite the cosines and sines as follows:

Substituting these exponential functions into the Trigonometric Fourier Series we can obtain the exponential Fourier Series.

The Trigonometric Fourier Series

Instead of solving the cn values using the integral we can solve them using the an and bn values from the Trigonometric Fourier Series by using the following equations.

Trigonometric to Exponential Conversion

Exponential to Trigonometric Conversion

Example 1

Find the exponential Fourier series of the sawtooth wave below.