Complex analysis is a classical branch of mathematics that studies properties of differentiable functions on the complex plane and has links to algebra, analysis, number theory and geometry. We start by describing the complex plane and by defining the notion of complex differentiability, which is a natural extension of the notion of differentiability of real functions to the complex plane. However, we will discover that complex differentiable functions are very rigid object which have many amazing properties. A key result covered in this course is Cauchy's Integral Theorem, which says that the integral of every complex differentiable function along a loop in the complex plane is zero. We will see many beautiful consequences of this result; for example, the Cauchy Integral formula, the Residue theorem and even a proof of the Fundamental theorem of algebra.

• Freitag/Busam: Funktionentheorie 1, Springer, 2000
• Reinhold Remmert, Georg Schumacher: Funktionentheorie 1. Springer, 2002
• Folkmar Bornemann: Funktionentheorie, Birkhäuser, 2012
• Wolfgang Fischer, Ingo Lieb: Funktionentheorie. Vieweg, Braunschweig 2003
• T. Needham: Anschauliche Funktionentheorie, Wissenschaftsverlag, 2001
• Elias Wegert: Visual Complex Functions, Birkhäuser, 2012
• Domain Coloring: http://www.visual.wegert.com/