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Section 10.6 Alternative Forms of Fourier Series

There are many different forms of the Fourier series and therefore different formulas for the Fourier series coefficients, depending on the physical context.

The factors that will help you choose the correct form of the Fourier series are:

  • The zeros of the trig functions sine and cosine occur when the argument is an integer multiple of \(2\pi\text{,}\) so you will see versions of the formulas both without and with factors of \(2\pi\) explicitly in the argument:

    \begin{equation} f(\theta) =\sum_{n=0}^{\infty} a_n\, \cos{n\theta} +\sum_{n=1}^{\infty} b_n\, \sin{n\theta}\tag{10.6.1} \end{equation}
    or
    \begin{equation} f(\theta) =\sum_{n=0}^{\infty} a_n\, \cos{2\pi n\theta} +\sum_{n=1}^{\infty} b_n\, \sin{2\pi n\theta}\tag{10.6.2} \end{equation}
    In the first case, the variable \(\theta\) runs from \(0\) to \(2\pi\) and in the second case, it runs from \(0\) to \(1\text{.}\)

  • The argument of special functions like sine, cosine, and exponential must always be dimensionless (see Section 7.8), so if the independent variable has dimensions, it must always be divided by some constant with the same dimensions. The most typical examples are: the independent variable is a (rectangular) spatial dimension like \(x\text{,}\) in which case it will appear in the formulas divided by a constant length, for example \(L\text{,}\) which represents the wavelength of the phenomenon

    \begin{equation} f(x) =\sum_{n=0}^{\infty} a_n\, \cos\left(\frac{2\pi nx}{L}\right) +\sum_{n=1}^{\infty} b_n\, \sin\left(\frac{2\pi nx}{L}\right)\tag{10.6.3} \end{equation}
    or the independent variable is time \(t\text{,}\) in which case it will appear in the formulas divided by a constant time, for example \(T\text{,}\) which represents the period of the phenomenon.
    \begin{equation} f(t) =\sum_{n=0}^{\infty} a_n\, \cos\left(\frac{2\pi nt}{T}\right) +\sum_{n=1}^{\infty} b_n\, \sin\left(\frac{2\pi nt}{T}\right)\tag{10.6.4} \end{equation}

  • The integration in the formulas must always be over the period of the phenomenon, e.g. \(L\) or \(T\) in the examples immediately above.

  • Using Euler's formula (see Section 2.5), sines and cosines can always be rewritten as (complex) exponentials. Therefore, there is an exponential form of the Fourier series.

    \begin{equation} f(\theta) =\sum_{n=-\infty}^{\infty} c_n\, e^{in\theta}\tag{10.6.5} \end{equation}
    A big advantage of the exponential form is that it is only necessary to use a single sum, although the sum begins at \(n=-\infty\) instead of \(0\) or \(1\text{.}\) The trade-off is that you are now in complex-number land. For a real-valued function, the constants \(c_{n}\) and \(c_{-n}\) must be complex conjugates of each other. (Why?) This exponential form can also come with and without various dimensions and with or without explicit factors of \(2\pi\) in the argument, as in the bullets above.

Activity 10.5. Formulas for the coefficients of different forms of Fourier series.

Starting from the formulas (10.5.2) and (10.5.4) for the coefficients for \(a_m\) and \(b_m\text{,}\) and using any method you know for changing variables in integrals, find formulas for the coefficients for each of the other forms of the Fourier series listed above.

Hint.
Don't forget to change the limits in the integrals appropriately. Also, notice that the overall constant factors change in each case.
Answer.
As an example, the formulas for the coefficients corresponding to the form (10.6.3) involving the spatial variable \(x\) are:
\begin{gather} a_n = \frac{2}{L} \int_0^L \cos\left(\frac{2\pi nx}{L}\right) f(x) \,dx\tag{10.6.6}\\ b_n = \frac{2}{L} \int_0^L \sin\left(\frac{2\pi nx}{L}\right) \,f(x)\,dx\tag{10.6.7} \end{gather}