**Title of the course:** Introduction to complex analysis, with a taste of hyperbolic geometry

**Instructor:** Dr. Alexandre Ramos-Peon

**Institution:** University of Oslo

**Dates:** 31 July – 13 August 2017

**Prerequisites:** Basic calculus and linear algebra.

**Level:** Graduate, advanced undergraduate, beginning undergraduate

**Abstract:** This two-week course is an introduction to complex analysis in one variable. The approach will be geometric in flavour,

and might be perceived as “lighter” than most very rigourous treatments, such as the one in Ahlfors. In the first week, we start from

scratch and develop notation for complex numbers, complex fucntions and their visualization, the complex derivative (Caudhy Riemann

equations and their interpretation, integral formulas and its consequences), as well as complex integration. An intuitive approach is prefered,

but we will do some analytical proofs.

In the second week, we study Möbius transformations on the complex plane and their geometry. The Riemann sphere is explored, and

Möbius transformations classified. Introduction to hyperbolic geometry: upper-half plane and disk models and their isometries. We finally

prove that isometries of the hyperbolic plane are exactly Möbius transformations.

**Textbook or/and course webpage:**

Conway “Functions of one complex variable “ (chapters I,III,IV, maybe V.3 and VI. 1)

Stein and Shakarchi “Complex Analysis” (chapters 1,2,8)

Needham “Visual introduction to complex analysis”

**Language: **EN