ANALYTIC FUNCTIONS | FMA280 |

**Aim**

The aim is to provide concepts and methods from real and complex analysis which are important for further studies within for example mathematics, economy, 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, in assimilating mathematical text and to communicate mathematics.

*Knowledge and understanding*

For a passing grade the student must

be able to explain the basic theory of analytic functions (derivatives and integrals).

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, and how the latter arise in connection with discrete systems.

be familiar with and have an understanding of basic concepts in the theory of time discrete systems.

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.

be able to show capability independently to choose appropriate methods to decide whether a function series can be differentiated or integrated term-wise, and furthermore be able to describe and interpret the convergence of such a series.

be able to show capability independently to choose appropriate methods to solve problems connected to analytic functions.

be able to show capability to choose appropriate methods to solve problem connected to complex integration.

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, absolute and conditional convergence. Function sequences and function series. Norms of functions and uniform convergence.

*Power series:* radius of convergence, integration and differentiation of power series, power series expansions of the elementary functions.

*Fourier series:* exponential and trigonometric Fourier series, questions of convergence, Parseval's formula.

*Analytic functions:* definition of analytic function, the Cauchy-Riemann equations. Elementary analytic functions. Cauchy's integral theorem and integral formula. Expansion in power series. The identity theorem. The residue theorem. Calculation of real integrals by residue calculus.

**Literature**

Spanne, S: System och transformer I. Tidsdiskreta system och komplex analys. KFS 2006.