# Fourier Analysis, 6 credits (TATA77)

Fourieranalys, 6 hp

### Main field of study

Mathematics Applied Mathematics Electrical Engineering Applied Physics Biomedical Engineering

First cycle

Programme course

Mats Aigner

### Director of studies or equivalent

Jesper Thorén
Course offered for Semester Period Timetable module Language Campus VOF
6CYYY Applied Physics and Electrical Engineering, M Sc in Engineering 5 (Autumn 2017) 1 1 Swedish Linköping o
6KMAT Mathematics 5 (Autumn 2017) 1 1 Swedish Linköping o
6CITE Information Technology, M Sc in Engineering 7 (Autumn 2017) 1 1 Swedish Linköping v

### Main field of study

Mathematics, Applied Mathematics, Electrical Engineering, Applied Physics, Biomedical Engineering

First cycle

G2X

### Course offered for

• Applied Physics and Electrical Engineering, M Sc in Engineering
• Mathematics
• Information Technology, M Sc in Engineering

### Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

### Prerequisites

Calculus (one and several variables), Linear Algebra and Complex analysis or equivalent.

### Intended learning outcomes

The course covers Fourier series as well as Fourier, Laplace and z-transforms in a unified treatment based on the foundations of distribution theory and complex analysis. It will give mathematical knowledge fundamental for treatment of problems in system engineering and physics. It is also a preparation for courses in partial differential equations. After a completed course, the student will be able to:

• Differentiate, integrate and transform distributions in one variable with particular emphasis on the Dirac distribution and its derivatives.
• Calculate Fourier series for simple periodic functions and distributions and determine convergence properties and estimate approximation errors in the mean.
• Solve linear differential equations with constant coefficients using distributions and Fourier- and Laplace transforms and linear difference equations using z-transforms.
• Using the complex inversion integral, in combination with residue calculus, to calculate inverse Laplace and z-transforms.

### Course content

Basic distribution theory in one variable. Basic properties of Fourier series, Fourier, Laplace and z-transforms. Convergence of Fourier series, point wise and in the mean. Parseval's formula. Integrals with a parameter. The Fourier transform. The inversion formula. Rules of manipulation. The convolution formula. Parseval's formula. Inversion formulas and their validity. Convolutions and their transforms. Transforms of distributions. Applications to engineering and science.

### Teaching and working methods

Lectures, problem classes.

### Examination

 TEN1 Written examination U, 3, 4, 5 6 credits

Four-grade scale, LiU, U, 3, 4, 5

### Department

Matematiska institutionen

Jesper Thorén

Mats Aigner

### Course website and other links

http://www.mai.liu.se/und/kurser/index-amne-tm.html

### Education components

Preliminär schemalagd tid: 62 h