Fourier Transform of Periodic Function with period a unless ka=2πn

Question

I'm currently going through a mathematical physics textbook question that claims:

If f(x) is periodic with period a then f ̃(k) = 0, unless ka = 2πn for integer n.

However, wouldn't the f(k) be a none zero term with a delta function? Or am I misinterpreting the question?

Answer 1

Not necessarily. $f(k)$ will be a sum of delta functions multiplied with some coefficients. Think of a simple example $f(x)=\sin(x)+\sin(2x)$