Complex Analysis, Numerical Analysis

Enrollement in the course requires an successful completion of the following courses:

Linear Algebra and its Applications (A0B01LAA)
Introduction to Calculus (A0B01MA1)

The course cannot be taken simultaneously with:

Mathematical Applications (A2B99MAA)

Lecturer:

Aleš Němeček, Jan Hamhalter (gar.)

Tutor:

Aleš Němeček, Jan Hamhalter (gar.)

Supervisor:

Department of Mathematics

Synopsis:

Introduction to complex variables: holomorphic functions, line integral and Cauchy's integral formula, power series expansions. Laurent series and residue theorem. Subsequent lectures are focused on the study of using the Maple for complex variable and computations in numerical analysis.

Requirements:

Syllabus of lectures:

1. Complex numbers and functions.

2. Holomorphic functions, Cauchy-Riemann equations.

3. Elementary and multivalued holomorphic functions.

4. Path integral. Cauchy's integral formula.

5. Power series expansions of holomorphic functions.

6. Laurent series. Classification of singularities.

7. Residue theorem and its applications.

8. Introduction to Maple.

9. Numerical Analysis: Approximation of functions, polynomial interpolation.

10. Error estimates for polynomial interpolation. Splines.

11. Numerical differentiation. Least square method.

12. Finding complex roots and solutions of systems of nonlinear equations.

13. Numerical solutions of systems linear a equations.

Syllabus of tutorials:

1. Complex numbers and functions.

2. Holomorphic functions, Cauchy-Riemann equations.

3. Elementary and multivalued holomorphic functions.

4. Path integral. Cauchy's integral formula.

5. Power series expansions of holomorphic functions.

6. Laurent series. Classification of singularities.

7. Residue theorem and its applications.

8. Introduction to Maple.

9. Numerical Analysis: Approximation of functions, polynomial interpolation.

10. Error estimates for polynomial interpolation. Splines.

11. Numerical differentiation. Least square method.

12. Finding complex roots and solutions of systems of nonlinear equations.

13. Numerical solution of systems of linear equations.

Study Objective:

Study materials:

1. Lang, S.: Complex Analysis, Springer, 1993.

2. Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T.: Numerical Recipes (The Art of Scientific Computing), Cambrige University Press, Cambrige, 1990.

3. Maple 13 User Manual, Maplesoft, Waterloo Maple Inc., 2009.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomT2:C4-459 Hamhalter J. Němeček A. 12:45–14:15 (lecture parallel1 parallel nr.101) Dejvice Laborator počítače
Tue
Fri	roomZ4:B2-362 Hamhalter J. Němeček A. 16:15–17:45 (lecture parallel1) Zikova ulice Pocitacova laborator K301
Thu	roomZ4:B2-362 Hamhalter J. Němeček A. 11:00–12:30 (lecture parallel1 parallel nr.101) Zikova ulice Pocitacova laborator K301
Fri

The course is a part of the following study plans: