# Complex Analysis

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
* 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 $\mathbb {R} ^{n}$ - (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)

### 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)

### Complex Analysis Part 2

* 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)

### Singularity and Residues - Part 3

* 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-automorphisms

* Riemann mapping theorem - (Wiki2Reveal slides)
** Schwarz's Lemma
** $\mathrm {Aut} (\mathbb {D} )$
* 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