MATH 419: Introduction to Complex Variables
Clayton Shonkwiler
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}