Complex Analysis

Introduction

Subject classification: this is a mathematics resource.

Course contains Wiki2Reveal Slides

Moving the argument of function $f$ in the complex number plane. The point $z$ has a blue color and $f(z)=\frac{1}{z}$ is marked in red color. $z$ is moved on a curve with $\gamma (t)=t\cdot e^{it}$.

Image of path in the complex numbers for the function $f(z)=\frac{1}{z}$

Complex analysis is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

Articles

• Algebra II
• Dummy variable
• Materials Science and Engineering/Equations/Quantum Mechanics

Slides for Lectures

Chapter 1 - Intoduction

• Complex Numbers - (Wiki2Reveal slides)
  • Heine-Borel Theorem
• Riemann sphere - (Wiki2Reveal slides)
• Exponentiation and roots - (Wiki2Reveal slides)

Chapter 2 - Topological Foundations

• Sequences and series - (Wiki2Reveal slides)
• Power series - (Wiki2Reveal slides)
• Topological algebra - (Wiki2Reveal slides)
• Topological space - Definition: Topology
• Norms, metrics, topology - (Wiki2Reveal slides)

Chapter 3 - Complex Derivative

• Holomorphic function - (Wiki2Reveal slides)
• Partial Derivative - (Wiki2Reveal slides)
• Cauchy-Riemann-Differential equation - (Wiki2Reveal slides)

• Application of Cauchy-Riemann Equations - (Wiki2Reveal slides)

Chapter 4 - Curves and Line Integrals

• Line integral in ${ℝ}^n$ - (Wiki2Reveal slides)
• Curves - (Wiki2Reveal slides)
  • Wikipedia: holomorphic function
  • Wikipedia:Integral
• Paths - (Wiki2Reveal slides)

• Path Integral - (Wiki2Reveal slides)

• Wikipedia: Curve integral
• Continuity and Limit of a sequence

• Trace of Curve - (Wiki2Reveal slides)

Chapter 5 - Holomorphic Functions

• Holomorphic function - (Wiki2Reveal slides)
  • Criteria - (slideset)
  • Wikipedia: Holomorphic function criteria
  • Differences from real differentiability
  • conformal mappings$(\ast )$,
  • Inequalities - (Wiki2Reveal slides)
  • rectifiable curve - (Wiki2Reveal slides)
• Curve Integral - (Wiki2Reveal slides)
• Path of Integration - (Wiki2Reveal slides)

• Goursat's Lemma (Details) - (Wiki2Reveal slides)
• Cauchy's Integral Theorem for Disks - (Wiki2Reveal slides)

• Identity Theorem - (Wiki2Reveal slides)
• Liouville's Theorem - (Wiki2Reveal slides)
• Representation with Taylor Series - (Wiki2Reveal slides)
  • Abel's lemma
  • Examples for Power Series
  • Approach for 1/z - (Wiki2Reveal slides)
  • Cauchy distribution - (Wiki2Reveal slides)
  • Logarithm - (Wiki2Reveal slides)
  • Normal distribution in Complex Analysis - (Wiki2Reveal slides)
  • Error function - (Wiki2Reveal slides)

Complex Analysis Part 2

• Chain - Wiki2Reveal slides)

• Cycle - (Wiki2Reveal slides)

• Laurent Series - (Wiki2Reveal slides)
  • Example Computation with Laurent Series - (Wiki2Reveal slides)

• Goursat's Lemma
• Cauchy Integral Theorem - (Wiki2Reveal slides)

• Cauchy's integral formula - (Wiki2Reveal slides)
• Example Computation with Laurent Series - (Wiki2Reveal slides)

• Maximum Principle - (Wiki2Reveal slides)

• Open Mapping (and Connectedness) Theorem - (Wiki2Reveal slides)

• Criteria for Holomorphy - (Wiki2Reveal slides)

Singularity and Residues - Part 3

• Winding number - (Wiki2Reveal slides)
• Singularities - (Wiki2Reveal slides)
• Example - exp(1/z)-essential singularity - (Wiki2Reveal slides)
• Residue - (Wiki2Reveal slides)
  • null-homologous
  • development in Laurent series,
  • Isolated singularity,
• Decomposition theorem,
• Casorati-Weierstrass theorem,
• Riemann's theorem on removable singularities
• Residue Theorem - (Wiki2Reveal slides)
• Real integrals with residue theorem
• Zeros and poles counting integral - (Wiki2Reveal slides)
• Rouché's theorem - (Wiki2Reveal slides)
• meromorphic function

Riemann mapping theorem-automorphisms

• Riemann mapping theorem - (Wiki2Reveal slides)
  • Schwarz's Lemma
  • $\mathrm{A}\mathrm{u}\mathrm{t}(𝔻)$
• analytical and harmonic function

Exercises

• Exercises for Introduction to Complex Analysis
• Sheet 1
• Sheet 2
  • Solution to Exercise 3
  • Solution to Exercise 4
• Sheet 3
• Sheet 4
• Sheet 5
• Paper 1
• Complex Analysis/Quiz

Lectures

• Cauchy-Riemann equations
• Cauchy Theorem for a triangle
• Complex analytic function
• Complex Numbers
• Divergent series
• Estimation lemma
• Fourier series
• Fourier transform
• Fourier transforms
• Laplace transform
• Riemann hypothesis
• The Real and Complex Number System
• Warping functions

Sample exams

Sample Midterm Exam 1 Sample Midterm Exam 2

See also

• Boundary Value Problems
• Introduction to Elasticity
• The Prime Sequence Problem
• Wikipedia: Complex analysis
• Complex number