## Lecture Notes

### Functions of a Complex Variable

Much part of hese lecture notes from Functions of a Complex Variable course I taught.

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

#### Problem Sets

#### References and Further Reading

Click on linked topics to view lecture notes.

Complex Algebra

Cauchy-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 for problems.

Problem Set 1: Complex Algebra

Problem Set 2: Cauchy-Riemann Conditions, Harmonic Functions

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

Problem Set 4: Taylor Series, Laurent Series

Problem Set 5: Residues, Cauchy's Residue Theorem

Problem Set 6: Applications of Residues

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

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

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