Monthly Archives: December 2014

A Tale of Two Integrals

Two integrals look almost the same, and even to those fairly well-versed in the art, are the same. But alas, as we shall see. Consider the integral $$I_1 = \int_0^{\pi} dx \frac{x \sin{x}}{1+\cos^2{x}} $$ This may be evaluated by subbing $x \mapsto \pi-x$ as follows: $$\begin{align} I_1 &= \int_0^{\pi} dx \frac{(\pi-x) \sin{x}}{1+\cos^2{x}} \\&= \pi \int_0^{\pi} […]

Weird integral whose simplicity is only apparent in the complex plane

Every so often we come across an integral that seems absolutely impossible on its face, but is easily attacked – in fact, is custom designed – for the residue theorem. I wonder why a first year complex analysis class doesn’t show off this case as Exhibit A in why the residue theorem is so useful. […]

Expansions of $e^x$

A very basic question was asked recently: What is a better approximation to $e^x$, the usual Taylor approximation, or a similar approximation involving $1/e^{-x}$? More precisely, given an integer $m$, which is a better approximation to $e^x$: $$f_1(x) = \sum_{k=0}^m \frac{x^k}{k!} $$ or $$f_2(x) = \frac1{\displaystyle \sum_{k=0}^m \frac{(-1)^k x^k}{k!}} $$ The answer is amazingly simple: […]