In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number i where i 2 = 1 . By utilising the Euler relation:

e i θ cos θ + i sin θ

we can replace the trigonometric functions by complex exponential functions. By also combining the Fourier coefficients a n and b n into a complex coefficient c n through

c n = 1 2 ( a n i b n )

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.


Learning Outcomes

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