**Summary:**

This is a topics course: There are some fixed elements, while other topics may vary from year to year and may even be adapted to the students needs and wishes while the course is in progress. Fixed are: Approximation theory; Normal families and Montel's theorem; Riemann mapping theorem; Product representations; Harmonic and subharmonic functions. A choice will be made of: Gamma, zeta and elliptic functions; Potential theory in the complex plane; Theory of Hardy spaces; Holomorphic dynamics in the complex plane; Entire functions of exponential type; Picard's theorem.

**Prerequisites:**

Elementary function theory, Real analysis, basic functional analysis, elementary Fourier analysis.

Chapter 1-6 and 8-11 of Rudin’s “real and complex analysis” are more than sufficient. One should be acquainted with the main concepts there. In particular, from real analysis we will occasionally use concepts and results concerning (complex) Borel measures, (complex) Riesz representation Theorem, Lp theory, Hilbert space theory and its application to Fourier series, Hahn Banach Theorems, Fubini’s Theorem, and Fourier transform (the inversion theorem and Plancherel).

**Target Group:**

Master Mathematics students with an interest in analysis. Also students with an interest in geometry (Riemann surfaces) may find this course useful.

**Take Home Exercises:**

**Take home exercises are meant as preparation for the exam. The students try to make the exercises at home, if they do not succeed, they should ask for help. Take home exercises are not graded, but a selection of the take home exercises will form the written exam. **

**Literature:**

**Syllabus Advanced function theory, This is permanently being adapted and still changing continuously,.****Additional: books by Conway, Narasimhan, Ahlfors, Garnett, Berenstein en Gay, Duren, Heins, Segal, Boas: an extended list is part of the notes.**

**Lecturer:**

Jan Wiegerinck