I am reading the chapter titled Fourier Analysis from Kreyszing's book "Advanced Engineering Mathematics".
There is a section which talks about changing the period from $$2\pi$$ to $$2L$$ and I am trying to understand how this "change of scale" takes place. I think this is a change of variables but I am unable to derive it by myself. Here is a snapshot of that section: https://pasteboard.co/oubtbmgJsS0r.png
I understand the need behind why you would want to do something like this but unable to do it myself.
For additional context, I have tried to do some manipulation myself on this function having period $p$ but without success : $$f(x+p) = f(x)$$
Can someone please help me with this change of variable?
Also, why is the value of $v$ in the textbook set to $\frac{\pi}{L}x$? What's the idea behind this?
Given that $f(x)$ has a period of $p$:
$$f(x+p) = f(x)$$
then $f\left(\frac{p}{2\pi}v\right)$, as a function of $v$, has a period of $2\pi$:
$$f\left(\frac{p}{2\pi}(v+2\pi)\right) = f\left(\frac{p}{2\pi}v+p\right) = f\left(\frac{p}{2\pi}v\right)$$