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Gauss’s Lemma is not a horrible combinatorial calculation

I’m not really sure why I’ve never seen this argument before, but it’s much cleaner and easier to digest than the arguments I’m used to that make Gauss’s Lemma look like it’s not part of algebra at all. Definition. A … Continue reading

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How to get GAP4 to describe a group

This is a command that’s tough to Google even if you know it exists.  To get GAP to describe a group with a common, human-readable name, you can use the StructureDescription command, as follows: gap> StructureDescription(Group([(1,3), (1,2,3,4)])); “D8” gap> StructureDescription(Group([(1,3), … Continue reading

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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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