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

Complex Analysis

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