The Complex Exponential Fourier Series

In chapter 9 we defined the trigonometric Fourier series for a periodic, piecewise continuous function. That was an infinite series of the form

${A}_{0}+{\displaystyle \sum _{k=1}^{\infty}\left[{a}_{k}\mathrm{cos}\left(2\pi {\omega}_{k}t\right)+{b}_{k}\mathrm{sin}\left(2\pi {\omega}_{k}t\right)\right]}.$

Dealing with this series can be somewhat tedious. Typically, for example, the constant term, the cosine terms and the sine terms must be computed separately.

In this chapter we will derive an alternative — the complex exponential Fourier series — which, basically, is just the trigonometric series rewritten in terms of complex exponentials. This may not seem to be much of an improvement, especially since it will require complex-valued functions in computations that, up to this point, have only involved real-valued functions. ...

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