Monthly Archives: December 2015

Systematic treatment of a deceptively messy Cauchy principal value integral

The problem here is to evaluate the following: $$PV \int_0^{\infty} dx \frac{\log^2{x}}{(x-1)^2(x-4) \sqrt{x}} $$ This can be done using complex analysis, but it is a pretty involved affair, deceptively so. After struggling with the problem of how to present the solution, I am going to lay out a systematic approach that ignores the motivation behind […]

Cauchy principal value of a convolution

The problem here is to compute the following convolution-type integral: $$\int_{-1}^1 dx \frac{\sqrt{1-x^2}}{\lambda-x}$$ When $-1 \lt \lambda \lt 1$, this integral is infinite, but its Cauchy principal value may be defined. This integral is interesting because of the branch points. And so, away we go… Consider the following contour integral: $$\oint_C dz \frac{\sqrt{z^2-1}}{\lambda-z} $$ where […]