I would like to calculate the Fourier Transformation of a Lorentz shaped peak. Removing all the physical constants, the integral boils down to:
$$\int_0^\infty \frac{\cos(2 \pi x)}{(x - b)^2 + a} dx$$
Under the assumption that $b = 0$ and using symmetry it may be transformed to:
$$\frac{1}{2}\int_\infty^\infty \frac{\cos(2 \pi x)}{x^2 + a} dx$$
After several steps I can evaluate it. But I am completely lost in the case where $ b \neq 0$. Could you give me some hints?