How do I evaluate
$$\int_{-\infty}^{\infty} \frac{e^{ix}}{x} dx$$
Tried several methods but could not succeed. Looks like the gamma function, but the lower limit isn't $0$.
This integral arrived while trying to compute the Fourier inverse of $\frac{1}{y^2-a^2}$ and then substituting $x=y\pm a$.
Thanks for your help.
The integral $\int_{-\infty}^\infty \frac{e^{ikx}}{x}\,dx$ does not converge as an improper Riemann integral due to the singularity at $x=0$.
However, the principal value, $\lim_{\epsilon\to0}\int_{|x|>\epsilon} \frac{e^{ikx}}{x}\,dx$, exists and is given by
$$\lim_{\epsilon\to0}\int_{|x|>\epsilon} \frac{e^{ikx}}{x}\,dx=\lim_{\epsilon\to0}\int_{|x|>\epsilon} \frac{\sin(kx)}{x}\,dx=\pi \text{sgn}(k)$$