# MAT-60106 Complex Analysis, 5 cr

#### Additional information

The course homepage is:

https://moodle.tuni.fi/

#### Person responsible

Petteri Laakkonen

#### Lessons

 Implementation Period Person responsible Requirements MAT-60106 2019-01 2 Petteri Laakkonen Final examination and exercise activity.

#### Learning Outcomes

After passing the course the student: - recognizes elementary functions and their properties and is able to solve equations consisting of elementary functions. - can decide when a function is analytic and knows the main features of such functions - is able to calculate complex integrals using the definition, the fundamental theorem of analysis, and Cauchy's integral theorem and formulas. - can find the Laurent's series for a given function,and knows when the series represent the original function. - can find zeros and poles of a function from its Laurent's series - can compute complex integrals using residues - can make logical conclusions, i.e., is able to make mathematical proofs

#### Content

 Content Core content Complementary knowledge Specialist knowledge 1. Complex numbers and elementary functions. Complex plane and its topology. Complex function. Applications 2. Continuous and differentiable functions. Analytical Fuctions. Cauchy-Riemann and Laplace equations. Applications: - Elliptic partial differential equations 3. Complex Integral. The fundamental theorem of analysis. Cauchy's integral theorem, Cauchy's integral formulas 4. Taylor's and Laurent's series. Residue. Applying residues to calculate real integrals.

#### Study material

 Type Name Author ISBN URL Additional information Examination material Book Complex Analysis for Mathematics and Engineering Mathews& Howell No Book Complex Variables and Applications Brown&Churchill 0-07-114065-4 No Book Introduction to Complex Analysis R. P. Agarwal, K. Perera, and S. Pinelas 978-1-4614-0195-7 No

#### Prerequisites

 Course Mandatory/Advisable Description MAT-01300 Insinöörimatematiikka X 3 Mandatory 1 MAT-01310 Insinöörimatematiikka A 3 Mandatory 1 MAT-01330 Insinöörimatematiikka C 3 Mandatory 1 MAT-01360 Matematiikka 3 Mandatory 1 MAT-01366 Mathematics 3 Mandatory 1 MAT-02100 Usean muuttujan funktiot Advisable 2 MAT-02106 Multivariable Calculus Advisable 2

1 . Required prerequisite knowledge.

2 . Recommended prerequisite knowledge.

Additional information about prerequisites
Prerequisite information consists of Engineering Mathematics 1-3 (15 cr), Mathematics 1-3 (15 cr), or any combination of courses with corresponding contents. In particular, basic knowledge of differentiation and integration of real functions is required. Knowledge on the basic contents of Multivariate calculus (formerly Mathematics 4), e.g., on partial differentiation, is recommended.

#### Correspondence of content

 Course Corresponds course Description MAT-60106 Complex Analysis, 5 cr MAT-60100 Complex Analysis, 5 cr MAT-60106 Complex Analysis, 5 cr MAT-31086 Complex Analysis, 5 cr

 Updated by: Laakkonen Petteri, 03.10.2019