Proofs

Topic	Year
Circular Lens Intersection Problem I came across this math problem on YouTube (see below), and took a stab at solving it. After viewing the associated StackExchange post, I decided the answers there were missing an analytical expression, so I solve for it here. In its most compact form, the solution is \((3 \theta - 1) x^2\), where \(L = 2x\) and \(\theta = \tan^{-1}\frac{1}{2}\).	2022
Fourier Transform Identity for an Infinite Impulse Train This is a classic signal processing problem / Fourier transform identity. Turns out the Fourier transform of an infinite Dirac delta impulse train is another infinite Dirac delta impulse train.	2015
Residues of Complex Functions Under a Linear Translation I wrote this proof to get some points back on a math exam in grad school. To me it was obvious (at the time) - residues of a complex-valued function remain constant under a linear translation. I prove this, and apply it to simplify the process of computing an inverse Laplace transform via the Bromwich integral.	2014

Topic

Year

Circular Lens Intersection Problem

I came across this math problem on YouTube (see below), and took a stab at solving it. After viewing the associated StackExchange post, I decided the answers there were missing an analytical expression, so I solve for it here. In its most compact form, the solution is \((3 \theta - 1) x^2\), where \(L = 2x\) and \(\theta = \tan^{-1}\frac{1}{2}\).

Shaded circle region

2022

Fourier Transform Identity for an Infinite Impulse Train

This is a classic signal processing problem / Fourier transform identity. Turns out the Fourier transform of an infinite Dirac delta impulse train is another infinite Dirac delta impulse train.

2015

Residues of Complex Functions Under a Linear Translation

I wrote this proof to get some points back on a math exam in grad school. To me it was obvious (at the time) - residues of a complex-valued function remain constant under a linear translation. I prove this, and apply it to simplify the process of computing an inverse Laplace transform via the Bromwich integral.

2014