COMPLEX ANALYSIS FMA037

Higher education credits: 6. Grading scale: TH. Level: G2 (First level). Language of instruction: The course will be given in Swedish. FMA037 overlap following cours/es: FMA013, FMA018, FMA030, FMA035, FMA280, FMA013, FMA018, FMA030, FMA035 och FMA280. Compulsory for: E2, F2. Optional for: D2, M3, N3, V4. Course coordinator: Director of Studies, Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: Basic courses in Analysis and Linear algebra. Assessment: Written test comprising theory and problems. Computer work. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

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

have knowledge of the definition and properties of the elementary analytic functions.

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 solve linear difference equations, and to carry out the solution essentially correct.

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.