Complex Analysis

Problem types

Chapters

1. Holomorphic functions – arithmetic, the logarithm, Cauchy-Riemann, power series, harmonic functions.

2. Cauchy theory – the integral theorems and what they immediately give.

3. Singularities – Laurent series, the classification, meromorphic functions.

4. Residues and contours – the residue theorem and the real integrals it evaluates.

5. Counting zeros – the argument principle, Rouché, Hurwitz.

6. Conformal maps – Möbius transformations, Schwarz, Riemann mapping.

Three pages collect material from several chapters:

• Review sheet – the statements of the chapters’ theorems, on one page.

• Counterexamples – functions showing that a theorem fails when one of its hypotheses is dropped, grouped by theorem.

• Standard integrals – real integrals evaluated by residues, with their contours and arc estimates.

Background and reference covers undergraduate analysis and standard series expansions; Appendix notes holds supplementary proofs and special functions.

The syllabus

Calculus and undergraduate analysis

• Continuity and differentiation in one and several real variables

• Inverse and implicit function theorems

• Compactness and connectedness in analysis

• Uniform convergence and uniform continuity

• Riemann integrals

• Contour integrals and Green’s theorem

Reference: Rudin.

Preliminary topics in complex analysis

• Complex arithmetic

• Analyticity, harmonic functions, and the Cauchy-Riemann equations

• Contour integration in \(\CC\)

References: Ahlfors ch. 1–2; Hille ch. 1, 2, 4; Stein–Shakarchi ch. 1.

Cauchy’s theorem and its consequences

• Cauchy’s theorem and integral formula

• Morera’s theorem

• Schwarz reflection

• Uniform convergence of analytic functions

• Taylor and Laurent expansions

• Maximum modulus principle and Schwarz’s lemma

• Liouville’s theorem and the fundamental theorem of algebra

• Residue theorem and applications

• Singularities and meromorphic functions

• Casorati-Weierstrass theorem

• Rouché’s theorem, the argument principle, and the open mapping theorem

• Estimates from the Cauchy integral formula: the Cauchy inequalities, and bounds on holomorphic functions and their derivatives on compact sets

References: Ahlfors ch. 4–6; Hille ch. 5, 7, 8, 9; Stein–Shakarchi ch. 2, 3, 5, 8 (§2–3).

Conformal mapping

• General properties of conformal mappings

• Analytic and mapping properties of linear fractional transformations

• Automorphisms of the disc, plane, and Riemann sphere

References: Ahlfors ch. 3, 8; Hille ch. 3, 4; Stein–Shakarchi ch. 8 (§1–2).

References

1. L. Ahlfors, Complex Analysis, third edition, McGraw-Hill.

2. E. Hille, Analytic Function Theory, vol. 1, Ginn and Company.

3. W. Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill.

4. E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.

Acknowledgements

Thanks to:

• Edward Azoff, for a problem sheet broken out by topic.

• Mentzelos Melistas, for explaining and documenting many solutions.

• Jingzhi Tie, for many additional problems and solutions. Notation used with those problems: \(f_x \coloneqq \dd{f}{x}\) for partial derivatives, and \(\bd_z^n f\) for \(f^{(n)}(z)\).

• Swaroop Hegde, for a number of proofs.