Monthly Archives: September 2015

Integral with two branch cuts

The problem is to evaluate the following integral: $$\int_{-1}^1 dx \frac{\log{(x+a)}}{(x+b) \sqrt{1-x^2}} $$ where $a \gt 1$ and $|b| \lt 1$. It should be obvious to those who spend time around these integrals that this integral does not converge as stated. However, we only have a simple pole at $x=-b$ so that we can compute […]

Fascinating Fourier Transform

Sometimes I come across a Fourier integral that I have no idea how to attack. And then I find that I can convert it to another, more familiar integral using complex analysis. So here’s an example of such a satisfying situation. The problem is to evaluate $$\int_{-\infty}^{\infty} dx \, (1+i a^2 x)^{-1/2} (1+i b^2 x)^{-1/2} […]