MATS121 Complex Analysis 1 (5 cr)
Algebraic and topological properties of complex numbers. Complex valued functions of one complex variable (polynomials, exponential function, trigonometric functions, logarithm). Complex differentiability, holomorphic (analytic) functions and their basic properties, contour integrals. Cauchy-Riemann equations. Local versions of the Cauchy integral theorem and integral formula. Liouville's theorem, maximum principle, fundamental theorem of algebra. (Freitag and Busam: Complex analysis, chapters 1 and 2)
After passing the course successfully the student:
- knows the algebraic and topological properties of complex numbers
- knows basic properties of complex functions
- knows the definition of a holomorphic function and knows their basic properties
- can use (and derive) Cauchy-Riemann equations and knows the connection between differentiability and Cauchy-Riemann equations
- can derive Cauchy integral theorem and integral formula for a disc and can apply them
- can prove the fundamental theorem of algebra
- can apply the theory of complex numbers in different areas of mathematics
Description of prerequisites
Vector analysis 1, Introduction to mathematical analysis 3 and 4.
- B.P. Palka: An Introduction to Complex Function Theory; ISBN: 0-387-97427-X
- Eberhard Freitag ja Rolf Busam: Complex analysis, toinen laitos, Universitext, Springer, 2009.
Teaching (5 cr)
Lectures 30 h, 8 sets of exercises