Lecturer

Prof. Dr. Paul Biran

Coordinator

Younghan Bae

Auxiliary Materials

The following material will be distributed during the final exam and you are not allowed to use your printed copy of this material. Also, it is not allowed to use your own auxiliary materials.

Auxiliary

Content

We will discuss the following subjects: Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, special functions, conformal mappings, Riemann mapping theorem.

week	material covered
1	Basic topology of the complex plane. Definitions from topology (open, closed, compact, path connected, continuity of a function).
2	Complex derivative functions and the Cauchy-Riemann equation. Comparing linear algebra over the real numbers and the complex numbers. The first order approximation of a complex function.
3	Consequences of the Cauchy-Riemann equation. A definition of path integrals.
4	Definition of a primitive function. Cauchy theorem for a holomorphic function on an unit circle.
5	Function of winding numbers. Global Cauchy theorem.
6	Cauchy estimate theorem. Maximum principle. Lemma of Schwarz.
7	A criterion when a continuous function has primitives. Homotopy between curves.
8	Convergence of a sequence of functions.
9	Taylor expansion. Radius of convergence. Convergence of a sequence of functions.
10	Laurent expansion.
11	Zeros of holomorphic functions. Singularities of holomorphic functions. Riemann extension theorem for removable singularity
12	Meromorphic functions. Casorati–Weierstrass theorem. Residue theorem. Argument principle.
13	Rouche's theorem. Branched covering. Open mapping theorem. Hurwitz theorem

Exercises

The new exercises will be posted here on every weekend. We expect you to look at the problems and to prepare questions for the exercise class on Tuesday.

Please hand in your solutions by the following Friday at 13:00 in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

Exercise classes

Literature

• "An introduction to complex function theory" by B. Palka (Undergraduate Texts in Mathematics. Springer-Verlag, 1991.)