Complex Analysis

Course guarantor:

Zdeněk Mihula

Lecturer:

Zdeněk Mihula

Tutor:

Ladislav Drážný, Zdeněk Mihula, Hana Turčinová

Supervisor:

Department of Mathematics

Synopsis:

The course is an introduction to the fundamentals of complex analysis and its applications. The basic principles of Fourier, Laplace, and Z-transform are explained, including their applications, particularly to solving differential and difference equations.

Requirements:

Syllabus of lectures:

1. Complex numbers. Limits and derivatives of complex functions.

2. Cauchy-Riemann conditions, holomorphic functions. Harmonic functions.

3. Elementary complex functions. Line integral.

4. Cauchy's theorem and Cauchy's integral formula.

5. Power series representation of holomorphic functions.

6. Laurent series. Isolated singularities.

7. Residues. Residue theorem.

8. Fourier series and basic properties of Fourier transform.

9. Inverse Fourier transform. Applications of Fourier transform.

10. Basic properties of Laplace transform.

11. Inverse Laplace transform. Applications of Laplace transform.

12. Basic properties of Z-transform.

13. Inverse Z-transform. Applications of Z-transform.

14. Spare lecture

Syllabus of tutorials:

1. Complex numbers. Limits and derivatives of complex functions.

2. Cauchy-Riemann conditions, holomorphic functions. Harmonic functions.

3. Elementary complex functions. Line integral.

4. Cauchy's theorem and Cauchy's integral formula.

5. Power series representation of holomorphic functions.

6. Laurent series. Isolated singularities.

7. Residues. Residue theorem.

8. Fourier series and basic properties of Fourier transform.

9. Inverse Fourier transform. Applications of Fourier transform.

10. Basic properties of Laplace transform.

11. Inverse Laplace transform. Applications of Laplace transform.

12. Basic properties of Z-transform.

13. Inverse Z-transform. Applications of Z-transform.

14. Spare tutorial

Study Objective:

Study materials:

[1] H. A. Priestley: Introduction to Complex Analysis, Oxford University Press, Oxford, 2003.

[2] E. Kreyszig: Advanced Engineering Mathematics, Wiley, Hoboken, 2011.

[3] L. Debnath, D. Bhatta: Integral Transforms and Their Applications, CRC Press, Boca Raton, 2015.

Note:

Further information:

https://moodle.fel.cvut.cz/courses/B0B01KANA

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Electronics and Communications 2018 (compulsory course in the program)
• Electrical Engineering, Power Engineering and Management (compulsory course in the program)
• Electrical Engineering, Power Engineering and Management - Applied Electrical Engineering 2018 (compulsory course in the program)
• Electrical Engineering, Power Engineering and Management - Electrical Engineering and Management (compulsory course in the program)