Quarter period symmetry in Fourier series

[physiks]

Suppose we have some function f(x) with period L. My book states that if it is even around the point x=L/4, it satisfies f(L/4-x)=-f(x-L/4), whilst if it is odd it satisfies f(L/4-x)=f(x-L/4). Then we define s=x-L/4 so we have for the function to be odd or even about L/4 that f(s)=±f(-s) respectively. I understand this.

Now the next part uses the fact that the coefficients for the sine terms in the Fourier series are given by
b_r=2/L∫f(x)sin(2πrx/L)dx for integer r, with the integral over one period L.
It then says that
b_r=2/L∫f(s)sin(2πrs/L+πr/L)ds.
It then goes on from here to show the certain conditions on the coefficients if f is even or odd about it's quarter period. However I don't understand this step. Surely x=s+L/4 so we can't just got from f(x) to f(s). However it seems this is essential to the whole proof. Thanks for any help!

See page 450 here (420 in the actual 'book'):
https://www.andrew.cmu.edu/user/gkesden/book.pdf
from the first new paragraph until the end of that same paragraph...

 

[physiks]

Anyone?