## Lecture Notes

### Functions of a Complex Variable

The intended purpose of these lecture notes is not in any way to attempt to provide in-depth discussions or any new insight on Functions of a Complex Variable but to provide beginners a quick crash course on basic ideas, compuational techniques, and applications of Functions of a Complex Variable so readers can advance more easily by filling in gaps with more in-depth knowledge from currently existing so many wonderful books on Functions of a Complex Variable.

These lecture notes should be accessible by undergraduate students of mathematics or physics who have taken Calculus, Multi-Variable Calculus, and preferrably also Linear Algebra.

#### Lecture Notes

#### Homework

#### References and Further Reading

Click on linked topics to view lecture notes.

Complex AlgebraCauchy-Riemann Conditions

Harmonic Functions

The Line Integral and Cauchy's Integral Theorem

Cauchy Integral Formula

Morera's Theorem

Cauchy's Inequality and Liouville's Theorem

Taylor Series

Laurent Series

Residues

Cauchy's Residue Theorem

More on Residues

The Three Types of Isolated Singularities

Residues at Poles

Zeros and Poles

Applications of Residues: Evaluation of Improper Integrals

Applications of Residues: Evaluation of Improper Integrals 2

Jordan's Lemma

Fresnel Integrals

Applications of Residues: Evaluations of Improper Integrals 3 (Indented Paths)

Analytic Continuation

Click on the following links topics for homework.

Homework 1: Complex Algebra

Homework 2: Cauchy-Riemann Conditions, Harmonic Functions

Homework 3: The Line Integrals, Cauchy Integral Theorem, Cauchy's Integral Fromula

Homework 4: Taylor Series, Laurent Series

Homework 5: Residues, Cauchy's Residue Theorem

Homework 6: Applications of Residues

Not in particular order. There are so many wonderful books on Functions of a Complex Variable. I have listed below only some of those books on Functions of a Complex Variable that I am familiar with.

Complex Analysis, Lars Ahlfors, 3rd Edition, McGraw-Hill, 1979

Complex Variables and Applications, James Brown and Ruel Churchill, 8th Edition, McGraw-Hill, 2008

Functions of One Complex Variable I, John B. Conway, 2nd Edition, Graduate Texts in Mathematics, Springer, 1978