Complex Analysis

Complex Analysis Syllabus

For the Oral Qualifying Exam

• Complex differentiation, Cauchy-Riemann equations, power series, exponential and trigometric functions.
• Cauchy's theorem and integral formula, Cauchy's inequalities, Liouville's theorem, Morera's theorem, classification of isolated singularities, Taylor series, meromorphic functions, Laurent series, fundamental theorem of algebra, residues, winding numbers, argument principle, Rouch\'e's theorem, local behaviour of analytic mappings, open mapping theorem.
• Harmonic functions, maximum principle, Poisson integral formula, mean value property.
• Conformal mappings, linear fractional transformations, Schwarz lemma.
• Infinite products, analytic continuation, multi-valued functions, Schwarz reflection principle, monodromy theorem.
• Statement and consequences of Riemann mapping theorem and Picard's theorem.

References

• L. Ahlfors, Complex Analysis
• J.B. Conway, Functions of One Complex Variable
• R. Churchill, Complex Variables and Applications
• S. Lang, Complex Analysis
• Levinson and Redheffer, Complex Variables
• K. Knopp, Theory of Functions, vols I-III.