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1988 IMO Problem 6

Problem 6 of the 1988 International Math Olympiad reads: Let a and b be positive integers such that a b + 1 divides .  Show that is the square of an integer. There’s a writeup of a solution at the … Continue reading

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The Poisson Summation Formula

There are various conventions for normalizing the Fourier transform. For the purposes of this post, let’s write The Poisson Summation Formula concerns the sum Where on Earth did that come from? Well, without regard to convergence issues, we have So … Continue reading

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Elliptic integrals, I

This has tripped me up in the past, so let’s write it down. Imagine we have an ellipse Let’s say we have some parametrization (x(t), y(t)) of the ellipse and we want to convert it into a unit-speed parametrization. We … Continue reading

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n-fold integral of the natural logarithm

The n-fold integral of the natural logarithm is given by where is the n-th harmonic number and C(z) is any polynomial of degree at most n-1.

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On the Gamma function

I want to say a bit about how the Gamma function can be characterized, because I’m not a huge fan of the ways I’ve seen in print. Let’s make this quick.  I want to know about functions γ such that: γ(1) … Continue reading

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Inner product spaces, I – Real closed fields

I’ve always felt sort of uneasy about the way linear algebra is presented: you start off doing all this stuff that makes complete sense, and works over arbitrary fields, and then suddenly you’re doing something with all these complex conjugates … Continue reading

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A more motivated proof of the Pythagorean theorem

So far every proof I’ve known of the Pythagorean theorem has adhered to a narrative along the lines of Notice, purely by accident, that in known right triangles it appears that the square on the hypotenuse is always equal to … Continue reading

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