# Complex functions, Small group teaching

## Extent

5 cr## Course dates

## Application period

## Fees

## Campus

## City

## Faculty or school

## Language of instruction

## Code

## Study fields

## Mode of study

## Study level

### Complex functions, 5 cr

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On the course students learn the basics results related to differentiation and integration of complex functions starting from the definition of complex numbers. After passing the course the student:

- is familiar with the complex numbers and their basic operations and can interpret them geometrically
- knows the basic functions and their properties and can solve complex equations involving them
- can decide when a function is analytic, e.g., by using the Cauchy-Riemann equations and knows the main features of analytic functions
- knows the definition of complex integrals and the related basic results, and can calculate complex integrals using them
- can find the Taylor’s or Laurent's series of a

given function and can study their convergence

knows what a singularity of a function is and can deduce its type

can make logical conclusions, e.g., is able to make mathematical proofs

concerning complex functions and explain why certain results may be used

**Core content**

- Complex numbers, basic operations and their

geometric interpretation. - Elementary complex functions and their

properties. - Continuity, differentiability and analyticity of

complex functions. Cauchy-Riemann equations. - Complex Integral and the related basic results:

The fundamental theorem of analysis, Cauchy-Goursat theorem, Cauchy's

integral theorem, Cauchy's integral formulas, and deformation theorem. - Taylor's and Laurent's series and their applications:

Singularities and Residue calculus.

**Complementary knowledge**

- Visualization of complex functions
- Applying residues to calculate real integrals.

**Specialist knowledge**

- Liouville’s theorem and the fundamental theorem

of algebra as a consequence of Cauchy’s integral formula - Applications: Harmonic functions and/or conformal

mappings

## Groups

### Group 1: Primetime-ryhmä / primetime group

09.03.2023 08:00 - 09:00

16.03.2023 08:00 - 09:00

23.03.2023 08:00 - 09:00

30.03.2023 08:00 - 09:00

13.04.2023 08:00 - 09:00

20.04.2023 08:00 - 09:00

27.04.2023 08:00 - 09:00

### Group 2: Primetime-ryhmä / primetime group

09.03.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 09:00 - 10:00, TAU Tietotalo TD308 opetustila (20)

### Group 3: Primetime-ryhmä / primetime group

09.03.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 10:00 - 11:00, TAU Tietotalo TD308 opetustila (20)

### Group 4: Primetime-ryhmä / primetime group

09.03.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 11:00 - 12:00, TAU Tietotalo TD308 opetustila (20)

### Group 5: Primetime-ryhmä / primetime group

09.03.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 12:00 - 13:00, TAU Tietotalo TD308 opetustila (20)

### Group 6: Primetime-ryhmä / primetime group

09.03.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 13:00 - 14:00, TAU Tietotalo TD308 opetustila (20)

### Group 7: Primetime-ryhmä / primetime group

09.03.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 14:00 - 15:00, TAU Tietotalo TD308 opetustila (20)

### Group 8: Primetime-ryhmä / primetime group

09.03.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

16.03.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

23.03.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

30.03.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

13.04.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

20.04.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

27.04.2023 15:00 - 16:00, TAU Tietotalo TD308 opetustila (20)

## Voluntary

### Exercise:

07.03.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

08.03.2023 12:00 - 14:00

14.03.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

15.03.2023 12:00 - 14:00

21.03.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

22.03.2023 12:00 - 14:00

28.03.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

29.03.2023 12:00 - 14:00

04.04.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

12.04.2023 12:00 - 14:00

18.04.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

19.04.2023 12:00 - 14:00

25.04.2023 14:00 - 16:00, TAU Konetalo K2108 opetustila (36)

26.04.2023 12:00 - 14:00

The course arrangements are based on the flipped learning methods meaning that the student first studies the weekly topics by himself/herself, solves weekly problems, and finally joins small group sessions where the weekly topics are revised and difficulties are discussed. The student can get help in solving the weekly problems by attending exercise sessions where the exercises are solved in small groups with the help of the teacher.

Assesment is based on the solved weekly exercises and small group exercises (70% weight) and the final exam (30% weight). An alternative method to pass the course is self studying and participating the exam. Students planning to study by themselves can attend the learning events if they want to.

The course materials are available in Finnish and in English. There are small groups session in Finnish and some in English, so choose the one that suits you. The exercise sessions are bilingual.

## Prerequisites

Prerequisite information consists of Differential and Integral Calculus, 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, e.g., on visualization of two-variable real functions and partial differentiation, is recommended.

## Compulsory Prerequisites

- Differential and Integral Calculus, MATH.APP.160, 5 cr
- Calculus, Part 2, MATH.MA.160, 5 cr

## Recommended Prerequisites

- Multivariable Calculus, MATH.APP.220, 5 cr

Course book and videos covering the course contents will be available. Below you can find optional material:

##### Material

- Type: Book
- Name: Introduction to Complex Analysis
- Author: R. P. Agarwal, K. Perera, and S. Pinelas
- Exam material: No
- ISBN: 978-1-4614-0195-7
- Language: English

##### Material

- Type: Book
- Name: Complex Analysis for Mathematics and Engineering
- Author: Mathews& Howell
- Exam material: No
- Language: English

##### Material

- Type: Book
- Name: Complex Variables and Applications
- Author: Brown&Churchill
- Exam material: No
- ISBN: 0-07-114065-4
- Language: English

The main course material is the text "Complex functions" by Petteri Laakkonen and Ville Koljonen and the related videomaterials. Additional learning material is the book "An Introduction to Complex Analysis" by Agarwal et

al.

"An Introduction to Complex Analysis" (Agarwal et al).

General scale, 0-5

Contact information

Email: open.studies.tau [at] tuni.fi

Phone: 0294 520 200

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