2 Fourier series implications

Since a sum of even functions is itself an even function it is not unreasonable to suggest that a Fourier series containing only cosine terms (and perhaps a constant term which can also be considered as an even function) can only represent an even periodic function. Similarly a series of sine terms (and no constant) can only represent an odd function. These results can readily be shown more formally using the expressions for the Fourier coefficients $a_n$ and $b_n$ .

Task!

Recall that for a $2\pi$ -periodic function

$b_n=\frac{1}{\pi }{\int \; <!--nolimits\; -->}_{-\pi }^{\pi }f\left(t\right)sinntdt$

If $f\left(t\right)$ is even, deduce whether the integrand is even or odd (or neither) and hence evaluate $b_n$ . Repeat for the Fourier coefficients $a_n$ .

Answer

It should be obvious that, for an odd function $f\left(t\right)$ ,

$a_n=\frac{1}{\pi }{\int \; <!--nolimits\; -->}_{-\pi }^{\pi }f\left(t\right)cosntdt=\frac{1}{\pi }{\int \; <!--nolimits\; -->}_{-\pi }^{\pi }\left(\text{odd function}\right)dt=0$

$b_n=\frac{2}{\pi }{\int \; <!--nolimits\; -->}_0^{\pi }f\left(t\right)sinntdt$

Analogous results hold for functions of any period, not necessarily $2\pi$ . For a periodic function which is neither even nor odd we can expect at least some of both the $a_n$ and $b_n$ to be non-zero. For example consider the square wave function:

Figure 17 :

This function is neither even nor odd and we have already seen in Section 23.2 that its Fourier series contains a constant $\left(\frac{1}{2}\right)$ and sine terms.

This result could be expected because we can write

$f\left(t\right)=\frac{1}{2}+g\left(t\right)$

where $g\left(t\right)$ is as shown:

Figure 18

Clearly $g\left(t\right)$ is odd and will contain only sine terms. The Fourier series are in fact

$f\left(t\right)=\frac{1}{2}+\frac{2}{\pi }\left(sint+\frac{1}{3}sin3t+\frac{1}{5}sin5t+\dots \right)$

and

$g\left(t\right)=\frac{2}{\pi }\left(sint+\frac{1}{3}sin3t+\frac{1}{5}sin5t+\dots \right)$

Task!

For each of the following functions deduce whether the corresponding Fourier series contains

1. sine terms only or cosine terms only or both
2. a constant term

   

Answer

Task!

Confirm the result obtained for the triangular wave, function 7 in the last Task, by finding the Fourier series fully. The function involved is

Answer