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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
br=2/L∫f(x)sin(2πrx/L)dx for integer r, with the integral over one period L.
It then says that
br=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...
Now the next part uses the fact that the coefficients for the sine terms in the Fourier series are given by
br=2/L∫f(x)sin(2πrx/L)dx for integer r, with the integral over one period L.
It then says that
br=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...
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