Complex functions, Small group teaching

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

Alternative learning events among themselves

Group 1: Primetime -ryhmä (suomenkielinen)

28.10.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 08:00 - 09:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 2: Primetime -ryhmä (suomenkielinen)

28.10.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 09:00 - 10:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 3: Primetime -ryhmä (suomenkielinen)

28.10.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 10:00 - 11:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 4: Primetime -group (Eglish language)

28.10.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 11:00 - 12:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 5: Primetime -ryhmä (suomenkielinen)

28.10.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 12:00 - 13:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 6: Primetime -group (Eglish language)

28.10.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 13:00 - 14:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 7: Primetime -ryhmä (suomenkielinen)

28.10.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 14:00 - 15:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 8: Primetime -group (Eglish language)

28.10.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
04.11.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
11.11.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
18.11.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
25.11.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
02.12.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)
09.12.2021 15:00 - 16:00, TAU Kampusareena A226 kokoushuone Teekkari (10)

Group 9: Valitse tämä, jos et pääse muihin primetime -ryhmiin / Select this if you cannot attend other primetime groups

25.10.2021 08:00 - 09:00
01.11.2021 08:00 - 09:00
08.11.2021 08:00 - 09:00
15.11.2021 08:00 - 09:00
22.11.2021 08:00 - 09:00
29.11.2021 08:00 - 09:00
06.12.2021 08:00 - 09:00

Other learning events

Exercise: Harjoitustilaisuus / Exercise session

26.10.2021 13:00 - 15:00
27.10.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
02.11.2021 13:00 - 15:00
03.11.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
09.11.2021 13:00 - 15:00
10.11.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
16.11.2021 13:00 - 15:00
17.11.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
23.11.2021 13:00 - 15:00
24.11.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
30.11.2021 13:00 - 15:00
01.12.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)
07.12.2021 13:00 - 15:00
08.12.2021 08:00 - 10:00, TAU Rakennustalo RH215 harjoitussali (70)

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).

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