Half-Range Expansions

In many physical problems the function $f(x)$ is only known over a finite interval, say $0 < x < L$. To express $f(x)$ as a Fourier series means that we need to extend the function to be valid over all $x$. If we choose to extend the function periodically, with period $L$, then we retrieve the results of Section 2.4.1. However, we could also extend the function in an even manner, as shown in Figure 2.5 to get a cosine series or as an odd function to get a sine series.

Figure 2.5: (a) A function defined on a interval $0 < x < L$, where $L=1$. (b) The even extension of the function onto the interval $-L < x < L$ is shown as a solid curve and the periodic extension of period $2L$ is shown as a dot-dash curve. (c) The odd extension of the function, solid curve, and the periodic extension of period $2L$, dot-dash curve.

The odd extension of $f(x)$ will generate a Fourier series that only involves sine terms and is called the sine half-range expansion. It is given by

$$f(x) =\sum_{n=1}^{\infty}b_{n} \sin {n \pi \over L}x,$$ (2.15)

where

$$\begin{eqnarray*} b_{n} & = & {2\over L} \int_{0}^{L} f(x) \sin {n\pi \over L}x dx, \end{eqnarray*}$$

and $n$ is a positive integer.

On the other hand, the even extension generates the cosine half-range expansion

$$f(x) = a_{0} + \sum_{n=1}^{\infty}a_{n} \cos {n \pi \over L}x,$$ (2.16)

where

$$\begin{eqnarray*} a_{0} & = & {1\over L} \int_{0}^{L}f(x) dx \\ a_{n} & = & {2\over L} \int_{0}^{L} f(x) \cos {n\pi \over L}x dx, \end{eqnarray*}$$

and $n$ is a positive integer.