TATA23 Transform Theory 2007

Study Guide@lith

Valid for year : 2007

TATA23	Transform Theory, 6 ECTS credits. /Transformteori/ For: Fys M
	Prel. scheduled hours: Rec. self-study hours: 160
	Area of Education: Science Subject area: Mathematics
	Advancement level (G1, G2, A): G2
	Aim: To give the student knowledge of the mathematical foundations of the transform methods, as well as the ability to apply these methods in problem solving. After completing this course, the student should be able to: cite and prove sufficient conditions for the existence of the different transforms, use and prove basic properties of the transforms, (e.g. behaviour at infinity, scaling and translation properties, rules for differentiation, integration, and multiplication by the independent variable), compute transforms of common functions, cite and use inversion theorems, uniqueness theorems, convolution formulas, Parseval's theorem, and Plancherel's theorem, apply transform methods to solve problems such as differential equations, difference equations, and convolution equations, cite and use some results about uniform convergence (continuity differentiability, and integrability of the limit function, Weierstrass M-test).
	Prerequisites: (valid for students admitted to programmes within which the course is offered) Linear Algebra and Calculus in several variables, Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
	Organisation: Teaching is done through lectures and problem classes.
	Course contents: The basic idea in this course is to study some important linear transforms, by means of which linear problems given by differential, integral and difference equations can be translated to more tractable linear algebraic problems, the solutions of which are possible to translate back to solutions of the original problems. The key to the usefulness of these methods to solve differential or difference equations is that differentiation as well as taking differences is translated to multiplication by the independent variable. The following is studied: Fourier series, which translate periodic functions to an infinite sequence of real or complex numbers, is used for analysis of periodic events. The problem of convergence is a central one, and uniform convergence, pointwise convergence and convergence in the mean for Fourier series is studied. Bessel's inequality and Parseval's theorem are key results. Fourier transforms, which translate functions defined on the real line to functions defined on the same interval, is used to study non-periodic events. The inversion formula is central and the tools also comprise rules of computation, the convolution formula and Parseval's theorem. The Laplace transform, which translates functions given on the real line to functions defined on the complex plane, is used for instance to solve initial value problems. The tools comprise rules of computation, the convolution theorem as well as the initial and end value theorem. The z-transform, which translates sequences of real or complex numbers to power series, is used for instance to solve difference equations. The tools are similar to the ones above.
	Course literature:
	Examination:
	Written examination	4 p	/	6 ECTS


Course language is Swedish. Department offering the course: MAI. Director of Studies: Göran Forsling Examiner: Link to the course homepage at the department Course Syllabus in Swedish

Linköping Institute of Technology
Contact: TFK , val@tfk.liu.se Last updated: 12/11/2007