Monthly Archives: November 2016

Computing an integral over an absolute value using Cauchy’s theorem

The problem is to compute the following integral: $$\int_{-1}^1 dx \frac{|x-y|^{\alpha}}{(1-x^2)^{(1+\alpha)/2}}$$ I will show how to compute this integral using Cauchy’s theorem. It was remarked that it should not be possible to use Cauchy’s theorem, as Cauchy’s theorem only applies to analytic functions, and an absolute value certainly does not qualify. True. Nevertheless, for the […]

Very nifty limit

Problem: Find the value of the limit $$\lim_{n \to \infty}n\left(\left(\int_0^1 \frac{1}{1+x^n}\,\mathrm{d}x\right)^n-\frac{1}{2}\right)$$ Solution: I chose this problem because the answer is highly nontrivial and just out of left field. But the process of getting there seems so straightforward; it is not really. Substitute $x=u^{1/n}$ in the integral and get $$I(n) = \int_0^1 \frac{dx}{1+x^n} = \frac1n \int_0^1 […]