A Result with Self-Inverse Functions

This post should really be Part 3, being a sequel to the series of posts about the Cauchy-Schlömilch transformation and the Glasser’s master theorem. However, I’ll frame this as a new fact and post. This post concerns functions that are inverses of each other, i.e., functions for which we have or for all . Functions

An Incredibly Overpowered Integration Technique – Part 2

In the previous post, we had derived the Cauchy-Schlömilch transformation, which is a wonderful substitution that eases the evaluation of a certain class of definite integrals. We reproduce the result here for convenience. Theorem (Cauchy-Schlömilch transformation): Let and assume that is a continuous function for which the integrals below are convergent and are construed as

An Incredibly Overpowered Integration Technique – Part 1

Today, we look at a beautiful transformation/substitution which is not presented in many textbooks and hence is often not part of the standard arsenal of tools for tackling integration problems. It is called the Cauchy-Schlömilch transformation “(after Cauchy who knew it by 1823, and the German mathematician Oskar Schlömilch (1823–1901) who popularized it in an

The Devious Sinc-Hole

The Borwein integrals are a mathematical oddity that manages to surprise even seasoned readers and hoarders of mathematical apocrypha. The unusual properties of the Borwein integrals were first presented in 2001 by the father-son duo of David and Jonathan Borwein and are a perfect example of the need for proof and rigor in math and

The Sophomore’s Dream

I get back to continue this journey into the world of curious integrals by telling you about the ‘Sophomore’s Dream’. The Sophomore’s Dream is the identity Discovered in 1697 by the Swiss mathematician Johann Bernoulli, this identity (or some minor variant of it) is today called the Sophomore’s Dream because it has a feel that

It’s Gettin’ Hot in Here

The inspiration for today’s post comes from physics, in particular, the theory of blackbody radiation. The motivation is to hopefully convince any of you reading this that sometimes some very interesting integrals (having deep connections to many fundamental problems in mathematics) have a way of raising their heads in problems in physics. Convinced? No? Let

A Curious Constant

In the pantheon of constants found in the world of mathematics, Catalan’s constant occupies a place of rightful reverence. This constant is the value of the infinite series or, in other words, the infinite sum of the reciprocals of the squares of the odd natural numbers with alternating signs. Precious little is known about the

Starting Off With a Slice of Pi

We choose to start this blog off with an integral that has a relationship with . Some digging tells me that this one first appeared in the Journal of the London Mathematical Society in 1944. Since then, it has popped up as A1 of the 1968 Putnam and also in IITJEE 2010, which is where