KindiakMath

Complex Analysis

Here are my write-ups on the theory of complex analysis and some non-trivial exercises therein. Complex analysis is one field of mathematics that, rather remarkably, ties many pieces of seemingly disparate mathematics together: from linear algebra and calculus to real analysis and topology, even touching base with aspects of abstract algebra and Fourier analysis. We also explore selected ideas in multivariable calculus as auxiliary tools for complex analysis.

Theory

Complex Differentiability

  1. The Square Root of -1
  2. Complex Differentiation
  3. Fréchet Derivatives
  4. Holomorphic Functions
  5. Extending the Logarithm

Complex Integrability

  1. The Complex Integral
  2. Cauchy-Goursat Theorem
  3. Cauchy’s Integral Formula
  4. Infinite Series Revisited
  5. Badly-Behaved Points
  6. Cauchy’s Residue Theorem

Baby Fourier Analysis

  1. Fourier Convergence
  2. The Fourier Transform
  3. Fourier Inversion
  4. Laplace Inversion

Exercises

Complex Differentiability

  1. Implicit Differentiation
  2. The Phasor Method
  3. The Jacobian
  4. Theoretical Optimisation

Complex Integrability

  1. Multivariate Line Integrals