Category Archives: Analysis

Weierstrass’s Smackdown of Dirichlet’s Principle

In 1856 Dirichlet made the following claim in a lecture:

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Escher’s Toroidal Print Gallery

If Art+Math brings one person to mind, it’s Escher. His tessellations present the best-known instance, but he did a lot more than that.

In April 2003, the mathematicians Bart de Smit and Hendrik Lenstra wrote a delightful article, Escher and the Droste effect, about Escher’s lithograph Prentententoonstelling. They pointed out that

We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers…

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Filed under Analysis, Geometry

Stirling’s Formula: Ahlfors’ Derivation

If you’re reading this blog, you probably know Stirling’s formula:

n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n

It’s not hard to estimate n! to within a factor of √2; I wrote up a note on this and even easier derivations. It’s quite a bit harder to show that the ratio of the two sides approaches a definite limit as n→∞ and that this limit is 1. You can find a variety of proofs in a number of places, one being Ahfors’ Complex Analysis.  I wrote up a note about this too, expanding on some of the details.

Incidentally, the two sides are asymptotic not just for positive integers n. Replace n! with Γ(z+1) on the left, and both n‘s with z‘s on the right. Allow z to go to infinity in the complex plane, while staying at least a fixed finite distance to the right of the imaginary axis. Then the two sides remain asymptotic. Ahfors proves this stronger result, and uses it to derive the integral form for the Γ function.

Note that if you replace the n‘s with z‘s, you have zz on the right. So you’ve got to worry about branches of the complex logarithm (since zz is defined as ez log z). The note deals with this (and other things).

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Filed under Analysis