1 Half-range Fourier series

So far we have shown how to represent given periodic functions by Fourier series. We now consider a slight variation on this theme which will be useful in HELM booklet  25 on solving Partial Differential Equations.

Suppose that instead of specifying a periodic function we begin with a function $f\left(t\right)$ defined only over a limited range of values of $t$ , say $0<t<\pi$ . Suppose further that we wish to represent this function, over $0<t<\pi$ , by a Fourier series. (This situation may seem a little artificial at this point, but this is precisely the situation that will arise in solving differential equations.)

To be specific, suppose we define   $f\left(t\right)=t^{2}0<t<\pi$

Figure 21

We shall consider the interval $0<t<\pi$ to be half a period of a $2\pi$ periodic function. We must therefore define $f\left(t\right)$ for $-\pi <t<0$ to complete the specification.

Task!

Complete the definition of the above function $f\left(t\right)=t^2,0<t<\pi$

by defining it over $-\pi <t<0$ such that the resulting functions will have a Fourier series containing

1. only cosine terms,
2. only sine terms,
3. both cosine and sine terms.

Answer

The point is that all three periodic functions $f_1\left(t\right),f_2\left(t\right),f_3\left(t\right)$ will give rise to a different Fourier series but all will represent the function $f\left(t\right)=t^2$ over $0<t<\pi$ . Fourier series obtained by extending functions in this sort of way are often referred to as half-range series.

Normally, in applications, we require either a Fourier Cosine series (so we would complete a definition as in (i) above to obtain an even periodic function) or a Fourier Sine series (for which, as in (ii) above, we need an odd periodic function.)

The above considerations apply equally well for a function defined over any interval.

Example 3

Obtain the half range Fourier Sine series to represent $f\left(t\right)=t^{2}0<t<3.$

Solution

We first extend $f\left(t\right)$ as an odd periodic function $F\left(t\right)$ of period 6 : $f\left(t\right)=-t^2,-3<t<0$

Figure 22

We now evaluate the Fourier series of $F\left(t\right)$ by standard techniques but take advantage of the symmetry and put $a_n=0,n=0,1,2,\dots$ .

Using the results for the Fourier Sine coefficients for period $T$ from HELM booklet  23.2 subsection 5,

$b_n=\frac{2}{T}{\int \; <!--nolimits\; -->}_{-\frac{T}{2}}^{\frac{T}{2}}F\left(t\right)sin\left(\frac{2n\pi t}{T}\right)dt,$

we put $T=6$ and, since the integrand is even (a product of 2 odd functions), we can write

$b_n=\frac{2}{3}{\int \; <!--nolimits\; -->}_0^{3}F\left(t\right)sin\left(\frac{2n\pi t}{6}\right)dt=\frac{2}{3}{\int \; <!--nolimits\; -->}_0^3{t}^{2}sin\left(\frac{n\pi t}{3}\right)dt.$

(Note that we always integrate over the originally defined range, in this case $0<t<3$ .)

We now have to integrate by parts (twice!)

So the required Fourier Sine series is

$F\left(t\right)=18\left(\frac{1}{\pi }-\frac{4}{{\pi }^3}\right)sin\left(\frac{\pi t}{3}\right)-\frac{18}{2\pi }sin\left(\frac{2\pi t}{3}\right)+18\left(\frac{1}{3\pi }-\frac{4}{27{\pi }^3}\right)sin\left(\pi t\right)-\dots$

Task!

Obtain a half-range Fourier Cosine series to represent the function

$f\left(t\right)=4-t0<t<4.$

First complete the definition to obtain an even periodic function $F\left(t\right)$ of period 8. Sketch $F\left(t\right)$ :

Answer

Now formulate the integral from which the Fourier coefficients $a_n$ can be calculated:

Answer

Now integrate by parts to obtain $a_n$ and also obtain $a_0$ :

Answer

Now write down the required Fourier series:

Answer

Note that the form of the Fourier series (a constant of 2 together with odd harmonic cosine terms) could be predicted if, in the sketch of $F\left(t\right)$ , we imagine raising the $t$ -axis by 2 units i.e. writing

$F\left(t\right)=2+G\left(t\right)$

Figure 23

Clearly $G\left(t\right)$ possesses half-period symmetry

$G\left(t+4\right)=-G\left(t\right)$

and hence its Fourier series must contain only odd harmonics.

Exercises

Obtain the half-range Fourier series specified for each of the following functions:

1. $f\left(t\right)=10\le t\le \pi \text{(sine series)}$
2. $f\left(t\right)=t0\le t\le 1\text{(sine series)}$
3. 1. $f\left(t\right)={\text{e}}^{2t}0\le t\le 1\text{(cosine series)}$
   2. $f\left(t\right)={\text{e}}^{2t}0\le t\le \pi \text{(sine series)}$
4. 1. $f\left(t\right)=sint0\le t\le \pi \text{(cosine series)}$
   2. $f\left(t\right)=sint0\le t\le \pi \text{(sine series)}$

Answer