Fourier Transforms and Sums

Question

Suppose I have the following sum: $$ \sum_{x = -\infty}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj $$ Assuming that everything is sufficiently smooth and convergent, then exchanging the sum with the integral gives: $$ \int_{-\pi}^{\pi} f(j) \; \sum_{x = -\infty}^{\infty} e^{i j x} dj = \int_{-\pi}^{\pi} f(j) \; 2 \pi \delta(j) \; dj = 2 \pi f(0) $$ Now suppose that instead I have half infinite sum: $$ \sum_{x = 1}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj $$ If $ f $ is even, ie. $ f(j) = f(-j) $ then I exploit this by symmetrizing and adding in $ x = 0 $ to find: $$ \sum_{x = 1}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj = \pi f(0) - \frac{1}{2} \int_{-\pi}^\pi f(j) dj $$ I'm stumped on the case if $ f $ is odd. I can't seem to find anything useful, which feels unbelievable. Thank you for any suggestions.