## MATH 419: Introduction to Complex Variables

Clayton Shonkwiler

shonkwiler.org

Motivating Question

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

### What does complex differentiability get you?

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

### What good is this?

• 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}

#### Complex Analysis

By Clayton Shonkwiler

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