References

Source of the practice problems

1.

Department of Mathematics, University of California at Riverside, Complex Analysis Qualifying Examinations, 2005–2025. The practice problems throughout these notes, and the whole of the chapter on problems beyond these notes, are taken from these papers. They are graduate examinations, sat before beginning doctoral research, and are reproduced here for study with the source acknowledged.

2.

The reading list set alongside those examinations — Ahlfors, Conway, Knopp, Narasimhan, Rudin, and Saks & Zygmund — is reproduced in full at the end of the chapter on problems beyond these notes, together with the syllabus the papers examine.

Further reading for the later chapters

1.

Ahlfors L.V., Complex Analysis, 3rd ed. McGraw–Hill. The standard reference for conformal mapping, the Riemann mapping theorem and harmonic functions.

2.

Forster O., Lectures on Riemann Surfaces, Springer. For Riemann–Roch, Riemann–Hurwitz, divisors and the sheaf-theoretic material, none of which is developed in these notes.

3.

Miranda R., Algebraic Curves and Riemann Surfaces, American Mathematical Society. A gentler route into the same material.

Prerequisite: Analytic Geometry and Calculus

Course Content

1.

The Complex Number System
The complex variable z; Cartesian and polar representation of z: Im z; Re z, \(|z|\), arg z, de moivre’s theorem; The complex plane; Lines and half lines in the complex plane.

2.

Elementary Theory of Power Series
Sequences; convergent, divergent and Cauchy Sequences; Series: convergent, divergent and absolutely convergent series; Uniform convergence. Power series; radius of convergence.

3.

Elementary Point Set Topology of \(\mathbb {C}\)
Neighbourhoods; open and closed point sets; Connectedness; continuous image of a connected set; regions.

4.

Analytic Functions
Limits and continuity; Differentiability at a point and in domain; the Cauchy-Riemann relations; Laplace’s equation; conjugate harmonic functions. Standard elementary functions: exp z , log z, the trigonometric and hyperbolic functions of z.

5.

Analytic Functions as Mappings
Paths and smooth paths; conformal mappings; Linear fractional transformations; cross ratio; symmetry; oriented circles.

6.

Complex Integration
Line integrals as functions of paths. Cauchy’s theorem; Cauchy integral formula; higher derivatives; Counting zeros; open mapping theorem. Morera’s theorem; Power series representation of analytic functions. Entire functions; fundamental theorem of calculus.

7.

Calculus of Residues
Classification of singular points. Taylor and Laurent series. Residue theorem evaluating of definite integrals. The argument principle; Rouche’s theorem.

8.

The Maximum Modulus Theorem
The Maximum principle. Schwarz’s Lemma and applications.

9.

Analytic Continuation
General analytic functions. Function element. Analytic continuation along a path. Homotopic curves; Monodromy theorem.