does integral_0^2 -sin(m pi x / 2) dx equal the negative kronecker delta_m,n, with n=0?

Question

The Kronecker delta_m,n function has several series definitions, one is
integral_0^2 sin(m pi x / 2) sin(n pi x / 2) dx. If we replace n=0 then this appears to sum to zero, but that makes no sense. The result should be a function that is one when m=0 and is zero otherwise. I cannot reconcile this contradiction, please help!

Answer 1

Paul B is right. since sin(0)=0, it's obvious that this formula (I assume that you meant for the upper limit to be 2 pi (2π) and not 2) gives you zero because the integrand is zero if either m or n or both are equal to zero. Thus it doesn't work for m=0 or n=0.

Answer 2

This definition is not valid for n=0