sarthakchatterjee1

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 . FunctionsContinue reading “A Result with Self-Inverse 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 asContinue reading “An Incredibly Overpowered Integration Technique – Part 2”

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 anContinue reading “An Incredibly Overpowered Integration Technique – Part 1”

Picks from The Putnam – Part 7

Over the past series of posts, I have to tried to highlight some beautiful integrals appearing on The Putnam. I will end this series today with Problem A3 from the 2016 Putnam, which is a wonderful marriage between functional equations and integration. Problem: Suppose that is a function from to such that for all realContinue reading “Picks from The Putnam – Part 7”

Picks from The Putnam – Part 6

The problem for this post is Problem A2 of the 1989 Putnam. The problem asks us to find the value of where . The region of integration is the rectangle . We divide this rectangle into two parts along the diagonal . Then, or, or, or,

Picks from The Putnam – Part 5

Today, we try and deal with Problem A5 from the 2005 Putnam. In particular, we are asked to evaluate I had dealt with this integral in a prehistoric Quora answer of mine (see: https://www.quora.com/What-are-some-interesting-integrals-requiring-creative-application-of-u-substitution/answer/%E0%A6%B8%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%95-%E0%A6%9A%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%9F%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%9C%E0%A7%80-Sarthak-Chatterjee). Today’s solution is a little different and will involve parametrizing the integral and then differentiating under the integral sign. Accordingly, letContinue reading “Picks from The Putnam – Part 5”

Picks from The Putnam – Part 4

Today, we look at Problem A2 from the 1992 Putnam. The problem goes as follows: Define to be the coefficient of in the power series expansion about of . Evaluate For starters, let us figure out what looks like. Indeed, and, so, or, and so, So our integral is just or,

Picks from The Putnam – Part 3

Today’s integral is Problem A3 from the 1997 Putnam. In particular, we are asked to evaluate This ominous-looking integral is not too hard once we realize how its different constituents fit. The infinite series in the left parenthesis is just Therefore, where we can justify the interchange of the sum and the integral by theContinue reading “Picks from The Putnam – Part 3”

Picks from The Putnam – Part 2

Problem B5 of the 1985 Putnam asks us to find the value of The problem also states that we can assume that We first make the substitution or, Our integral then becomes or, or, Now, let so that Thus, This is just a dummy variable, and so we can revert to so that We nowContinue reading “Picks from The Putnam – Part 2”

Picks from The Putnam – Part 1

The William Lowell Putnam Mathematical Competition, an undergraduate intercollegiate math competition often abbreviated to “The Putnam”, has had some beautiful integrals show up in it. We are today going to try and evaluate Problem B1 from the 1987 Putnam, which asks us to find the value of We use here the fact that So carryingContinue reading “Picks from The Putnam – Part 1”