More on the Kronecker Delta and Fourier Series

Modified my spreadsheet to sum the integral. So here is \int_{-\pi}^{\pi}sin(mx)sin(ny) and sin(mx)sin(nx) with m = n . (Integral is the dark orange line)

sin(x)sin9y) with integral and x = y

And for n = 5m

sin(x)sin(y) with integral when y = 5x

And for n = 2.1m :

sin(x)sin(y) with integral where y = 2.1x

So, it’s pretty clear \delta_{mn} (the Kronecker delta) applies when both m and n are integers.