I want to split $ J= \int_0^{\infty} e^{ik_ax}f(x) dx$ into a real and imaginary part as follows:
$ J= \int_0^{\infty} e^{ik_ax}f(x) dx = \frac{1}{2} \widetilde{f}(k_a) +\frac{1}{2\pi i} \mathcal{P}\int_{-\infty}^{\infty} \frac{\widetilde{f}(k)}{k-k_a} dk $ , where $\mathcal{P}$ denotes the Cauchy value of the integral and $\widetilde{f}(k) $ is defined as the Fourier transform of $ f(x)$: $\widetilde{f}(k)= \int_{-\infty}^{\infty} e^{ikx} f(x) dx $.
Does this identity have a name and how would one proof it?