### Clayton Shonkwiler PRO

Mathematician and artist

Clayton Shonkwiler

**Motivating Question**

What does it mean for a complex function to be differentiable, and what does differentiability get you?

- Complex-differentiable functions are analytic
- Complex-differentiable functions are harmonic
- Complex-differentiable functions can make some cool pictures

- Analytic number theory and combinatorics
- Prime number theorem, Riemann Hypothesis, L-functions, etc.

- Fourier/Laplace analysis
- PDEs, probability theory, etc.

- Complex geometry
- Riemann surfaces, elliptic curves, algebraic geometry, complex manifolds
- Differential geometry, including minimal surfaces

- Science and engineering
- Fluid dynamics, quantum mechanics, signal processing, etc.

0 = \lim_{N \to \infty} \int_{\gamma_N} \frac{\pi \cot(\pi z)}{z^2} dz \\
= \sum_{n=-\infty}^\infty \operatorname{Res}\left(\frac{\pi \cot(\pi z)}{z^2}, n \right) = 2\sum_{n=1}^\infty \frac{1}{n^2} - \frac{\pi^2}{3}

By Clayton Shonkwiler

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