This gives me the impression that you are confusing Fourier series with Fourier transforms.

If a function is periodic, with period T, then it has a Fourierseries, in which the n-th (complex) Fourier coefficient is .

Fouriertransformsare for functions (non-periodic) that are integrable over the whole real line, and the Fourier transform is a function, defined by .

For the two functions f_1 and f_2, the integrals for the Fourier transforms are

,

.