MATC21 Complex Analysis 1, 7.5 credits, is an alternative-compulsory course at upper-basic level for a Bachelor/Master of Science degree in mathematics. The course can be taken as an stand-alone course. The course is given at half-study pace during the second half of each autumn semester. The course is given in English.

Course Content
The course treats basic theory of analytic functions:

Elementary properties of analytic functions in one variable. Complex differentiability and Cauchy-Riemann equations. Calculation rules. Elementary examples of analytic functions: power series expansions, exponential functions, branches of logarithms, and functions defined by these calculation rules.
Contour integrals in the complex plane. Cauchy’s integral theorem and integral formula. Existence of a primitive function and local power series expansion of analytic functions. Cauchy estimates, Liouville’s theorem, and the fundamental theorem of algebra.
Theory of meromorphic functions: singular points, Laurent series expansion and the residue theorem. Residue calculus.
Further elements of the theory of holomorphic functions such as argument principle, Rouché’s theorem, and open mapping property.
Harmonic functions. Regularity, existence of harmonic conjugate, mean value property, maximum principle, Poisson integrals

Teaching
The teaching consists of lectures and seminars. The seminars are based on selected exercises and require active student participation, including preparation.

Assessment
The examination consists of a written examination and an oral examination at the end of the course. The oral examination is only given for students who have passed the written examination.

Course Literature

T. W. Gamelin: Complex analysis, Undergraduate Texts in Mathematics, Springer, 2001

Official Course Description

Course Evaluation

Course evaluation reports are published at https://www.maths.lu.se/english/education/courses/course-evaluations/