Entry requirements

120 credit points including Complex Analysis and Topology I

Learning outcomes

In order to pass the course (grade 3) the student should

know the Weierstrass factorisation theorem for entire functions and the Mittag-Leffler theorem for meromorphic functions;

be able to give an account of the concept of normal family;

be able to outline a proof of the Riemann mapping theorem;

be able to give an account of the basic properties of harmonic functions, Poisson's formula and the principle of Harnack;

know Jensen's formula and how to apply it;

be familiar with the concept of analytic continuation and the monodromy theorem;

be able to outline the construction of a modular function and the proof of the "little Picard theorem";

be familiar with the basic properties of subharmonic functions, and know how to use subharmonicity for the study of Dirichlet's proble;

be able to solve problems within the area of the course and to give proofs of central theorems.

Content

Infinite series and products. Partial fractions and factorisation. The gamma and the beta functions, Stirling's formula, the method of steepest descent. Riemann's zeta function. Normal families, the Riemann mapping theorem. Harmonic functions, Poisson's formula, Jensen's formula, the distribution of zeros of entire functions. Analytic continuation: continuation along arcs, the monodromy theorem, the modular function and Picard's theorem. Subharmonic functions. Dirichlet's problem.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.