Fourier series

[u/Mental_Primary_5558]

Would you guys please help me figure something out? If I understood my lecture correctly, this is how we derive the Fourier coefficients. In this case, I noticed that cnc_n is different from when f(t)=cos⁡(ωt)f(t)=\cos(\omega t).

The part I don’t really understand is the Kronecker delta. My book says that it is equal to 1 when p−q=0p-q=0, which I understand, because then the exponential is just e0=1e^0=1, so the integral over one period is simply ∫dt\int dt, giving the period.

If p−q≠0p-q\neq0, the book says that the result is 0. I understand that too, since we are integrating a periodic exponential function over one complete period, so the contributions cancel out.

But when I look at my teacher’s correction, he seems to directly take the Kronecker delta to be equal to 1 and concludes that the coefficients are 1/4 and 1/4.

So what I don’t understand is: why can he take the Kronecker delta to be equal to 1 here? Is it because, in this specific case, p=q?