Entry requirements

120 credit points Mathematics including Complex Analysis and Real Analysis.

Learning outcomes

In order to pass the course the student should be able to

• give an account for the fundamental concepts of holomorphic, meromorphic and harmonic functions as well and use these concepts in solving concrete problems;
• give an account for the concept of a normal family;
• give an account for the concepts of analytic continuation and monodromy and how to use them in concrete situations;
• give an account for the concepts of Riemann surfaces and complex manifolds and how to construct the Riemann surfaces of multivalued functions;
• give an account for the construction of a modular function;
• apply the central theorems such as Riemann's mapping theorem and Picard's little theorem in the relevant contexts.

Content

The space of holomorphic functions. Meromorphic functions. Hurwitz's theorem. Weierstrass's factorisation theorem. Mittag-Leffler's theorem for meromorphic functions. The complec projective space. Implicit function theorem. Infinite series and products. Partial fraction expansion and factorisation of holomorpic functions. The Gamma function. Riemann's zeta function. Normal families. Local geometry of holomorphic functions. Automorphisms. Riemann's mapping theorem. Harmonic functions. Poisson's formula and Dirichlet's problem. Jensen's formula. Subharmonic functions. Distribution of zeros of entire functions. Analytic continuation. The monodromy theorem. Complex manifolds and Riemann surfaces. Modular function and Picard's little theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written assignments given during the course.