The course is devoted to the theory of analytic functions of one complex variable.
Prerequisites
Real analysis, including uniform convergence and uniform continuity. Acquaintance with Green’s formula will be a benefit, but it is not indispensable.
Contents of the course
- Complex numbers and the complex plane.
- Definition of holomorphic functions, elementary
properties. Cauchy–Riemann equations. - Conformal mappings (explicit examples).
- Contour integration.
- Index of a closed path w.r.t. a point.
- The Cauchy package.
- Laurent series and isolated singularities.
- Analytic continuation.
- Geometric aspects of the theory: argument principle, open mapping principle, maximum modulus principle, Schwarz lemma.
- Residues. Evaluating integrals using residues.
- Riemann’s mapping theorem, with proof.
- (If time permits.) Christoffel–Schwarz theorem.
- (If time permits.) Elliptic functions.
Grading policy
The numerical grade equals 0.3*(homework)+0.3*(midterm exam)+0.4*(final exam). The midterm and final exams are written, open-book.
Textbooks
Textbooks on complex analysis abound. Below we mention just three of them, not meaning that others are any worse.
- John B. Conway. Functions of One Complex Variable. Springer-Verlag, New York-Berlin, 1978.
- Theodore W. Gamelin. Complex analysis. Springer-Verlag, New York, 2001.
- Serge Lang. Complex Analysis. Springer-Verlag, New York, 1999.