COMPLEX ANALYSIS | FMA037 |

**Aim**

The aim is to provide concepts and methods from complex analysis which are important for further studies within for example mathematics, physics, field theory, mathematical statistics, control theory, signal theory, and for professional work in the future. The aim is also to make the students develop their ability in problem solving and in assimilating mathematical text.

*Knowledge and understanding*

For a passing grade the student must

be able to show an understanding of the concept of convergence of a series, och be familiar with and be able to use some criteria to decide convergence.

be able to show an understanding of how functions and signals can be represented in different ways, as sequences and as function series.

have some experience and understanding of mathematical and numerical software.

*Skills and abilities*

For a passing grade the student must

be able to show capability independently to choose appropriate methods to decide whether a numerical series converges or diverges, and in the case of convergence to estimate its sum with different methods.

be able to show good capability to identify situations where different kinds of Fourier expansions are suitable, and to choose appropriate methods to derive such expansions.

in connection with problem solving be able to show capability to integrate knowledge from the different parts of the course.

with proper terminology, in a well structured way and with clear logic be able to explain the solution to a problem.

**Contents**

*Sums and series:*Sequences, recursive equations, numerical series, power series, Fourier series.*Complex elementary functions:*polynomials, rational functions, exponential and logarithmic functions.*Complex differentiation:*definition of analytic function, the Cauchy-Riemann equations.*Complex integration:*Cauchy's integral theorem and integral formula. Power series expansions of analytic functions.

**Literature**

Spanne, S: Konkret analys. KF-Sigma 1995.