# Complex Analysis

Structure Type: | Study unit |
---|---|

Code: | IX00BE87 |

Curriculum: | TT 2023 |

Level: | Bachelor of Engineering |

Year of Study: | 3 (2025-2026) |

Semester: | Spring |

Credits: | 5 cr |

Responsible Teacher: | Mäkelä, Jarmo |

Language of Instruction: | Finnish |

Taking the course in advance? See the courses during the academic year 2023-2024.

## Learning Outcomes

In the Complex Analysis course the student learns to differentiate and integrate functions of a complex variable. Using complex analysis one may, for instance, evaluate integrals, which could not be evaluated by any other means. One of the applications of complex analysis is the general theory of integral transforms that are used when solving linear differential equations frequently encountered in engineering. It turns out that if the function to be solved from a differential equation is replaced by its integral transform, the equation to be solved simplifies considerably.

## Contents

1) Complex numbers and their functions,

2) Analytic functions: The Cauchy-Riemann equations,

3) Line integrals in the complex plane: The Cauchy integral theorem,

4) Power series of complex functions,

5) Poles and residues of complex functions,

6) The Cauchy theorem of residues,

7) Evaluation of definite integrals by means of residues,

8) Fourier series,

9) Fourier transforms,

10) Solution of differential equations by means of Fourier transforms,

11) Laplace transforms,

12) Solution of differential equations and their systems by means of Laplace transforms,

13) Z-transform,

14) Difference equations and their solution by means of Z-transforms,

15) The Green function method.

## Recommended or Required Reading and Other Learning Resources/Tools

Suggested Reading: E. Kreyszig: Advanced Engineering Mathematics (Wiley).