In audio signal processing, the exponential Fourier series is essential for synthesizing sounds. For instance, a complex musical note can be decomposed into simpler sinusoidal waves, each with a unique frequency and amplitude.

The exponential fourier series presents periodic signals as the sum of complex exponentials at positive and negative harmonic frequencies.

Euler's identity is applied to expand the exponential term into its cosine and sine components. This is substituted back into the Fourier series.

The coefficients for each term in the series are calculated by integrating over one period of the function.

Upon substituting back into the series, a concise representation of the function in terms of the complex exponential is obtained.

The three forms of the Fourier series - Sine-Cosine Form, Amplitude-Phase Form, and Complex Exponential Form - are all interconnected.

To illustrate, consider a square wave signal. Through the Exponential Fourier Series, this square wave can be depicted as a sum of sinusoids, each with a frequency that is an odd multiple of the fundamental frequency and an amplitude inversely proportional to its frequency.