Entry requirements

120 credits including 90 credits in mathematics. Participation in Real Analysis and Complex Analysis. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• give an account for the fundamental concepts of holomorphic and meromorphic functions,
• reconstruct holomorphic and meromorphic functions with given zeros and singularities,
• give an account for the fundamental properties of subharmonic functions,
• 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 covering spaces,
• construct Riemann surfaces of multivalued functions.

Content

The space of holomorphic functions. Subharmonic functions. Meromorphic functions. Weierstrass's factorisation theorem. Mittag-Leffler's theorem for meromorphic functions. The Gamma function. Riemann's zeta function. Jensen's formula. Distribution of zeros of entire functions. Analytic continuation. The monodromy theorem. Riemann's mapping theorem. Covering spaces and Riemann surfaces. Holomorphic mappings and Picard's theorems. A brief theory of elliptic functions.

Instruction

Lectures.

Assessment

Written assignments (8 credits) and written project work (2 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.