# Fourier series change of variable

1. Jul 4, 2006

### Benny

Hi, I'm working on the (odd) square wave function

$$f\left( t \right) = \left\{ {\begin{array}{*{20}c} { - 1, - \frac{T}{2} \le t < 0} \\ { + 1,0 \le t < \frac{T}{2}} \\ \end{array}} \right\}$$

The question says to move the origin of t to the centre of an interval in which f(t) = +1 (ie. consider the even square wave function). A part of the question says to express the square wave function as a cosine series by making the change of variable $t' = t - \frac{T}{4}$.

Calculate the Fourier coefficients involved by making the suggested change of variables in the result 12.8

Result 12.8 is:

$$f\left( t \right) = \frac{4}{\pi }\left( {\sin \omega t + \frac{{\sin 3\omega t}}{3} + \frac{{\sin 5\omega t}}{5} + ...} \right)...(12.8)$$

This isn't apart of result 12.8 but the sine Fourier coefficients are:

$$b_r = \frac{2}{{\pi r}}\left[ {1 - \left( { - 1} \right)^r } \right]$$

I don't know how to calculate the coefficients for the cosine series using (12.8) and the change of variables t' = t - (T/4). Using the sin(A+B) expansion for each sine term in result 12.8 is tedious so should I write f(t) as follows and then use the sine addition formula on the general sine term?

$$f\left( t \right) = \sum\limits_{r = 1}^\infty {\frac{2}{{\pi r}}\left[ {1 - \left( { - 1} \right)^r } \right]\sin \left( {\frac{{2\pi rt}}{T}} \right)}$$

I probably haven't analysed the information carefully enough but I'm not sure how to start this. I've only just started on Fourier series. Any help would be good thanks.

Last edited: Jul 5, 2006