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