Author Archives: Ron

Real integral evaluation via the residue theorem with two branch points and a log-squared term

Every so often there comes an integral that I see as a major teaching opportunity in complex integration applications. This integral represents one such opportunity. Problem: Evaluate the definite integral $$\int_0^{\infty} dx \, \frac{\log^2{x} \, \log{(1+x)}}{1+x^2} $$ This integral may be evaluated using the residue theorem. The analysis involves two branch points with respective branch […]

A simple but nifty inequality

Problem: Given $f:[0,1] \to \mathbb{R}$ is integrable over $[0,1]$, and that $$\int_0^1 dx \, f(x) = \int_0^1 dx \, x f(x) = 1$$ show that $$\int_0^1 dx \, f(x)^2 \ge 4$$ The way to the solution here is not trivial. I started by always recognizing that, with integral inequalities, it never hurts to start with […]

Another crazy integral, another clinic on using the Residue Theorem

The problem posed here is to show that $$ \int_0^{\infty} dx \, \frac{\coth^2{x}-1}{\displaystyle (\coth{x}-x)^2+\frac{\pi^2}{4}} = \frac{4}{5} $$ The OP actually seemed to know what he was doing but could not get the correct residue that would allow him to get the stated result. In showing a link to Wolfram Alpha, the OP revealed that he […]

Another integral that Mathematica cannot do

The integral to evaluate is $$\int_0^{\infty} dx \frac{\sin{\left (\pi x^2 \right )}}{\sinh^2{(\pi x)}} \cosh{(\pi x)}$$ Given the trig functions in the integrand, it makes sense to use the residue theorem based on a complex integral around a rectangular contour. As has been my experience with these integrals, the integrand of the complex integral will not […]

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 […]

Integral of polynomial times rational function of trig function over multiple periods

Problem: Compute $$\int_0^{12\pi} dx \frac{x}{6+\cos 8x}$$ It looks like the OP is trying to compute the antiderivative and use the fundamental theorem of calculus. With multiple periods, that approach is paved with all sorts of difficulty that is really an artifice related to the functional form of the antiderivative. In truth, there should be no […]

A method of evaluating a double integral that nobody taught you in school

Many times we are given integrals to evaluate. The standard way to evaluate is to find a series of transformations that will render the integral into something we know how to evaluate and then proceed. Examples of such transformations are substitutions, parts, replacement of an integrand with another integral, reversing order of integration, and so […]

Deceptively easy product

The problem is to evaluate $$\prod_{n=2}^{\infty} \left (1+\frac{(-1)^{n-1}}{a_n} \right ) $$ where $$a_n = n! \sum_{k=1}^{n-1} \frac{(-1)^{k-1}}{k!} $$ This is one of those cases where trying out a bunch of numbers really isn’t going to help all that much. The $a_n$ look kind of like $n!$, except off by some. Even so, I can find […]

A sum I can only imagine being evaluated using the residue theorem

The challenge this time is to evaluate the following sum: $$\sum_{n=1}^{\infty} \frac{1}{n^3\sin{\left (\sqrt{2} \, n\pi \right)}}$$ NB (20 Nov 2016) I just got word that Wolfram fixed the problem described below. See below for details. It should be clear that the sum converges…right? No? Then how do we show this? Numerical experiments are more or […]

A little perspective on reaching 100K Rep on Math.SE

I stumbled onto on December 16, 2012. Before that, I was occasionally on the prowl for Putnam exam prep questions, various university math contests, and stuff like the IBM monthly puzzle.  At this point 1132 days later, I forget what I was looking for in the first place – likely a new problem to […]

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 […]

Computing the Convolution of Two Pulses: Graphical vs Analytical

Recently, a user on Math.SE presented a problem of computing the convolution of two pulses: a triangular pulse (impulse response) $$h(t) = \begin{cases} t & 0 \lt t \lt 2 T \\ 0 & t \lt 0 \cup t \gt 2 T\end{cases} $$ and a rectangular pulse (input) $$x(t) = \begin{cases} 1 & 0 \lt […]

Integral with two branch cuts II

The problem here is to compute $$\int_0^\infty \log(1+tx)t^{-p-1}dt$$ where $p\in(0,1)$ and $x>0$. This is a great problem for contour integration. Just tricky enough to be really interesting. What makes it interesting is that there are two functions in the integrand needing their own separate branch cuts. One must keep in mind that each function only […]

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} […]

Cauchy Principal Value

On Math.SE, we frequently get integrals that simply do not exist in the usual sense because the integration path intersects a pole of the integrand. Occasionally, we come across such integrals in the course of evaluating integrals of functions with removable singularities using complex methods. However, sometimes such integrals are interesting in their own right. […]

Inverse Laplace Transforms and Delta Functions

Problem: find the inverse Laplace transform of $$F(s) = \frac{s}{s-1}$$ Solution: Well, this one should be easy and one wonders why we are even bothering.  Just split $F$ up as follows: $$F(s) = \frac1{s-1} + 1$$ The ILT of $1/(s-1)$ is simply $e^t$, and the ILT of $1$ is $\delta(t)$.  Done. Or are we?  Why […]

Adventures in integration, University Edition

I got a request from an old friend whom I didn’t even know existed. He is the son of my grand-advisor, if that makes any sense. He teaches, among others, a course in Real Analysis at a university in Australia. I know I must like this guy because he says stuff like this: Currently, within the School […]

Inverse Laplace Transform with a coinciding pole and branch point

Recently, the following Laplace transform was asked to be inverted: $$F(s) = s^{-a-1} e^{-s^a} $$ where $a \in (0,1)$. This is a tough problem for two reasons. One is that there is very little chance of there being an analytical result for arbitrary values of $a$. The other, however, is more subtle: there is a […]

The art of using the Residue Theorem in evaluating definite integrals

As many of you know, using the Residue Theorem to evaluate a definite integral involves not only choosing a contour over which to integrate a function, but also choosing a function as the integrand. Many times, this is an easy task when integrating, say, rational functions over the real line. Sometimes there are less trivial […]