Fourier Series Overview

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Section 17.2 Fourier Series Overview

The basic idea of a Fourier series is that any (piecewise smooth) periodic function can be accurately represented by a (possibly infinite) sum of sine and cosine functions whose period is an integer multiple of the period of the function. The formula for the Fourier series for the \(f(\theta)\) with period \(2\pi\) (i.e. \(f(\theta+2\pi)=f(\theta)\)) is

\begin{equation} f(\theta) = \frac12 a_0 + \sum_{n=1}^\infty a_n \cos n\theta + \sum_{n=1}^\infty b_n \sin n\theta\tag{17.2.1} \end{equation}

The extra factor of \(\frac12\) in the \(a_0\) term is a convention. Note, there is no \(b_0\) term since \(\sin{0}\) is identically zero.

The coefficients \(a_n\) and \(b_n\) are uniquely determined by \(f(\theta)\text{.}\) The formulas are

\begin{align} a_n \amp= \frac{1}{\pi}\int_0^{2\pi} \cos (nx)\, f(x) \,dx\notag\\ b_n \amp= \frac{1}{\pi}\int_0^{2\pi} \sin (nx)\, f(x) \,dx\notag \end{align}

The derivation of the coefficients for a given \(f(\theta)\) can be found in Section 17.4. The derivation is straightforward and completely analogous to finding coefficients for vectors that are arrows in space. It is also analogous to many such calculations that you will do with other, more abstract vector spaces, so it is well worth your time to work through. The geometric idea behind the formulas for the coefficients can be found in Section 14.5, Section 14.6, an Section 14.7.

In applied settings, you will want formulas for Fourier series involving variables with dimensions. Alternative versions can be found in Section 17.5. Most often, a finite number of terms in the Fourier series are used to approximate a periodic function. You can explore how this works in Section 17.7, Section 17.8, an Section 17.10. In cases with high symmetry, you may be able to simplify your calculations by exploiting the symmetries of the harmonic functions, see Section 17.11.