In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number where . By utilising the Euler relation:
we can replace the trigonometric functions by complex exponential functions. By also combining the Fourier coefficients and into a complex coefficient through
we find that, for a given periodic signal, both sets of constants can be found in one operation.
We also obtain Parseval’s theorem which has important applications in electrical engineering.
The complex formulation of a Fourier series is an important precursor of the Fourier transform which attempts to Fourier analyse non-periodic functions.
- know how to obtain a Fourier series
- be competent working with the complex numbers
- be familiar with the relation between the exponential function and the trigonometric functions
- express a periodic function in terms of its Fourier series in complex form
- understand Parseval’s theorem
2 Revision of the exponential form of a complex number
3 Properties of the complex Fourier coefficients
4 Parseval’s theorem
4.1 Proof of Parseval’s theorem.