Complex Analysis

Subject classification: this is a mathematics resource.

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

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

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

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

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

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

• Line integral in ${\displaystyle \mathbb {R} ^{n}}$ - (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)

• Holomorphic function - (Wiki2Reveal slides)
  • Criteria - (slideset)
  • Wikipedia: Holomorphic function criteria
  • Differences from real differentiability
  • conformal mappings${\displaystyle (\ast )}$,
  • 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)

• Chain - Wiki2Reveal slides)

• Cycle - (Wiki2Reveal slides)

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

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

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

• Winding number - (Wiki2Reveal slides)
• Singularities - (Wiki2Reveal slides)
• Example - exp(1/z)-essential singularity - (Wiki2Reveal slides)
• Residuals - (Wiki2Reveal slides)
• null-homologous
• development in Laurent series,
• Isolated singularity,
• decomposition theorem,
• Casorati-Weierstrass theorem,
• Riemann Removability Theorem
• 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 - (Wiki2Reveal slides)
  • Schwarz's Lemma
  • ${\displaystyle \mathrm {Aut} (\mathbb {D} )}$
• analytical and harmonic function

• 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

• 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 Midterm Exam 1 Sample Midterm Exam 2

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