Asymptotical Methods
Code  Completion  Credits  Range  Language 

01ASY  Z,ZK  3  2+1  Czech 
 Lecturer:
 Jiří Mikyška (guarantor)
 Tutor:
 Jiří Mikyška (guarantor)
 Supervisor:
 Department of Mathematics
 Synopsis:

Examples. Addition parts of mathematical analysis (generalized Lebesgue integral, parametric integrals.) Asymptotic relations a expansions  properties; algebraical and analytical operations. Applied asymptotics of sequences and sums; integrals of Laplace and Fourier type.
 Requirements:

Basic courses of Calculus (in the extent of the courses 01MA1, 01MAA24 held at the FNSPE
CTU in Prague).
 Syllabus of lectures:

1. Landau symbols
2. Asymptotic sequences and Asymptotic expansions of functions.
3. Basic properties of asymptotic expansions and algebraic operations with them
4. Differentiation and integration of the asymptotic relations
5. Asymptotics of sequences
6. Asymptotics of series
7. Asymptotics of the roots of algebraic equations
8. Supplement to mathematical analysis  generalized Lebesgue integral
9. Asymptotics of the Laplace integrals, Laplace theorem, Watson's lemma.
10. Examples, applications of the asymptotic methods.
 Syllabus of tutorials:

1. Examples of the asymptotic expansions of functions and their properties
2. Basic properties of asymptotic expansions and algebraic operations with them
3. Asymptotics of sequences, Stirling's formula
4. Asymptotics of series, approximation of pi.
5. Asymptotics of the roots of algebraic equations
6. Asymptotics of the Laplace integrals, applications of the Laplace theorem and Watson's lemma
7. Examples, applications of the asymptotic methods.
 Study Objective:

EulerMaclaurin summation formula, perturbation methods, Laplace method, Watson's lemma.
Skills: Application of the asymptotical methods for investigation of the asymptotics of sequences, series, and integrals of Laplace and Fourier type.
 Study materials:

Key references:
[1] P. D. Miller: Applied Asymptotic Analysis, Graduate Studies in Applied Mathematics, Vol. 75, American Mathematical Society, 2006.
Recommended references:
[2] E. T. Copson: Asymptotic Expansions, Cambridge University Press, 1965.
[3] N. G. de Bruin: Asymptotic Methods in Analysis, North Holland Publishing Co., 1958.
[4] F. J. Olver: Asymptotics and special functions, Academic press, New York (1974)
 Note:
 Timetable for winter semester 2021/2022:
 Timetable is not available yet
 Timetable for summer semester 2021/2022:
 Timetable is not available yet
 The course is a part of the following study plans:

 Matematické inženýrství (compulsory course of the specialization)
 Matematická fyzika (elective course)
 Aplikované matematickostochastické metody (elective course)