Complex Analysis

## Introduction

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| Subject classification: this is a mathematics resource.

|

Course contains Wiki2Reveal Slides Moving the argument of function  f {\displaystyle f} in the complex number plane. The point  z {\displaystyle z} has a blue color and  f ( z ) = 1 z {\displaystyle f(z)={\frac {1}{z}}} is marked in red color.  z {\displaystyle z} is moved on a curve with  γ ( t ) = t ⋅ e i t {\displaystyle \gamma (t)=t\cdot e^{it}} . Image of path in the complex numbers for the function  f ( z ) = 1 z {\displaystyle 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

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

## Slides for Lectures

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### Chapter 1 - Intoduction

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

### Chapter 2 - Topological Foundations

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

### Chapter 3 - Complex Derivative

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* 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

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* Line integral in  R n {\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)

### Chapter 5 - Holomorphic Functions

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* 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)

### Complex Analysis Part 2

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* 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

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* 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

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

## Exercises

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* 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

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* 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

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Sample Midterm Exam 1 Sample Midterm Exam 2

## See also

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