Selected Topics in Mathematics

Garant předmětu:

Jiří Mikyška

Lecturer:

Jiří Mikyška

Tutor:

Jiří Mikyška

Supervisor:

Department of Mathematics

Synopsis:

Fourier series: complete orthogonal systems, expansion of functions into Fourier series, trigonometric Fourier series and their convergence. Complex analysis: derivative of holomorphic functions, integral, Cauchy's theorem, Cauchy's integral formula, singularities, Laurent series, residue theorem.

Requirements:

Basic Calculus (in the extent of the courses 01MA1, 01MAA2-3, or 01MAB2-3 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Theory of Fourier series in a general Hilbert space, complete orthogonal systems, Bessel inequality, Parseval equality.

2. Fourier series in L2, trigonometric system, Fourier coefficients, Bessel inequality, Parseval equality, expansion of a function into trigonometric series.

3. Criteria of convergence of Fourier series.

4. Analysis of complex functions: derivative, analytical functions, Cauchy-Riemann conditions.

5. Contour integral of complex functions of a complex variable, theorem of Cauchy, Cauchy's integral formula.

6. Expansion of an analytic function into a power series, isolated singularities, Laurent expansion, residue theorem.

Syllabus of tutorials:

1. Summary of properties of function series, investigation of the uniform convergence of function series.

2. Fourier series in a general Hilbert space, Gramm-Schmidt ortogonalization, ortogonal polynomials.

3. Trigonometric system in L2. Expansions of trigonometric functions into trigonometric Fourier series, investigation of convergence of the trigonometric series. Summation of some series using the Fourier expansions.

4. Elementary functions of complex variables: polynomials, exponential function, goniometric functions, complex logarithm

5. Analysis in a complex domain: continuity, derivative, Cauchy-Riemann conditions.

6. Evaluation of contour integrals of complex functions of a complex variable, applications of the Cauchy theorem, Cauchy integral formula and residue theorem.

Study Objective:

Expansion of functions to the Fourier series and investigation of their convergence, application of theory of analytic functions for evaluation of curve integrals in complex plane and evaluation of some types of definite integral of real functions of a real variable.

Skills: to use expansions of functions into a Fourier series to evaluate sums of some series, evaluation of definite integrals using the theory of functions of complex variable.

Study materials:

Key references:

[1] J. Dunning-Davies, Mathematical Methods for Mathematicians, Physical Scientists and Engineers, John Wiley and Sons Inc., 1982.

Recommended references:

[2] A. S. Cakmak, J. F. Botha, and W. G. Gray, Computational and Applied Mathematics for Engineering Analysis, Springer-Verlag Berlin, Heidelberg, 1987.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• BS Matematické inženýrství - Aplikované matematicko-stochastické metody (compulsory elective course)
• BS Informatická fyzika (compulsory elective course)
• BS jaderné inženýrství B (compulsory course of the specialization)
• BS Dozimetrie a aplikace ionizujícího záření (compulsory elective course)
• BS Experimentální jaderná a částicová fyzika (compulsory elective course)
• BS Inženýrství pevných látek (compulsory elective course)
• BS Diagnostika materiálů (compulsory elective course)
• BS Fyzika a technika termojaderné fúze (compulsory elective course)
• BS Fyzikální elektronika (compulsory elective course)