Fourier series for arbitary functions

Question

I am unable to understand transition from Fourier series of function with $2\pi$ period to Fourier series of function with arbitrary period, am referring advance engineering mathematics by Erwin here is image of that part in book

Answer 1

Suppose $f$ has periodic $2p > 0$. Then $f(p x/\pi)$ has period $2\pi$. So \begin{align} f(px/\pi)& \sim \frac{1}{2\pi}\int_{-\pi}^{\pi}f(py/\pi)dy \\ &+\sum_{n=1}^{\infty}\frac{1}{\pi}\int_{-\pi}^{\pi}f(py/\pi)\sin(ny)dx\sin(nx) \\ &+\sum_{n=1}^{\infty}\frac{1}{\pi}\int_{-\pi}^{\pi}f(py/\pi)\cos(ny)dy\cos(nx). \end{align} Let $w=py/\pi$ so that $dy=(\pi/p) dw$ in the integrals and then let $z=px/\pi$ in $f(px/\pi),\sin(nx),\cos(nx)$ to obtain

\begin{align} f(z) &\sim \frac{1}{2p}\int_{-p}^{p}f(w)dw \\ &+\sum_{n=1}^{\infty}\frac{1}{p}\int_{-p}^{p}f(w)\sin(n\pi w/p)dw\sin(n\pi z/p) \\ &+\sum_{n=1}^{\infty}\frac{1}{p}\int_{-p}^{p}f(w)\cos(n\pi w/p)dw \cos(n\pi z/p). \end{align}