Often times, I run into an equation that I don't know how to derive. Naturally, I spend time trying to derive it myself, or finding some other resource that explains the derivation. But sometimes, the resources I find aren't satisfactory for one reason or another: too narrow in scope, too hand-wavy, too bogged down in mathematical rigor, or too broad to be easily applicable. Basically, if I can't find a resource that provides a derivation satisfying my own personal taste, I'll put it up on here. If you find any mistakes, please send them to me (see the contact form in the about section).
In this derivation, I seek out the derivation for Stirling's approximation for large factorials, using principles from asymptotic analysis, and furthermore, I try and expand the derivation beyond the usual first or second orders typically provided.
In this derivation, I try to figure out where the factor of $2\pi$ comes from in the inverse Fourier transform. Turns out (at least doing it my way), it's a lot harder than it seems.