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MTH304Sciences3 Unitsintermediate

Complex Analysis I

This course introduces students to the fundamental concepts of complex analysis. It covers complex numbers, functions, analytic functions, limits, continuity, Taylor and Laurent series, and bilinear transformations. The course aims to equip students with the skills to investigate geometry on the complex plane, solve problems using Cauchy's integral formula and Liouville's Theorem, and apply complex analysis to various mathematical and engineering problems.

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150h
Study Time
13
Weeks
12h
Per Week
advanced
Math Level
Course Keywords
Complex NumbersAnalytic FunctionsTaylor SeriesLaurent SeriesBilinear Transformation

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
∑
Math Level
Advanced Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Complex Numbers

2

Complex Functions

3

Analytic Functions

4

Limits and Continuity

5

Taylor Series

6

Laurent Series

7

Bilinear Transformations

8

Cauchy-Riemann Equations

9

Contour Integration

10

Residue Theorem

Total Topics10 topics

Requirements

Knowledge and skills recommended for success

Calculus I

Calculus II

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assignments

Comprehensive evaluation of course material understanding

Written Assessment

end-of-course examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

Explore the career paths this course opens up for you

Mathematician

Apply your skills in this growing field

Engineer

Apply your skills in this growing field

Data Analyst

Apply your skills in this growing field

Financial Analyst

Apply your skills in this growing field

Software Developer

Apply your skills in this growing field

Industry Applications

Real-world sectors where you can apply your knowledge

EngineeringPhysicsComputer ScienceFinanceTelecommunications

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1:

8h

Unit 1: Complex Numbers

4 study hours
  • •Review the definition of complex numbers and their arithmetic operations.
  • •Practice plotting complex numbers on the complex plane.
  • •Solve problems involving modulus, conjugate, and polar coordinates of complex numbers.

Unit 2: Complex Functions

4 study hours
  • •Study the definition of complex functions and their properties.
  • •Examine examples of complex functions and their transformations.
  • •Practice finding the real and imaginary parts of complex functions.
Week
2

Module 2:

5h

Unit 1: Analytic Functions

5 study hours
  • •Understand the definition of analytic functions and their properties.
  • •Study the Cauchy-Riemann equations and their applications.
  • •Practice determining whether a given function is analytic.
Week
3

Module 2:

5h

Unit 2: Limit and Continuity

5 study hours
  • •Define the concepts of limits and continuity for complex functions.
  • •Study the properties of limits and continuous functions.
  • •Practice solving problems involving limits and continuity of complex functions.
Week
4

Module 3:

6h

Unit 1: Taylor and Laurent Series

6 study hours
  • •Study the definition and properties of Taylor series.
  • •Learn how to find the Taylor series expansion of a function.
  • •Practice solving problems involving Taylor series expansions.
Week
5

Module 3:

6h

Unit 1: Taylor and Laurent Series

6 study hours
  • •Understand the definition and properties of Laurent series.
  • •Learn how to find the Laurent series expansion of a function.
  • •Practice solving problems involving Laurent series expansions.
Week
6

Module 4:

7h

Unit 1: Bilinear Transformation

7 study hours
  • •Study the definition and properties of bilinear transformations.
  • •Learn how to apply bilinear transformations to solve problems.
  • •Practice solving problems involving bilinear transformations.
Week
7

Module 1:

8h

Unit 1: Complex Numbers

4 study hours
  • •Review complex numbers, functions, and their properties.
  • •Work through practice problems on complex arithmetic and transformations.

Unit 2: Complex Functions

4 study hours
  • •Review complex functions and their properties.
  • •Work through practice problems on complex functions and transformations.
Week
8

Module 2:

5h

Unit 1: Analytic Functions

5 study hours
  • •Review analytic functions and the Cauchy-Riemann equations.
  • •Work through practice problems on determining analyticity.
Week
9

Module 2:

5h

Unit 2: Limit and Continuity

5 study hours
  • •Review limits and continuity of complex functions.
  • •Work through practice problems on limits and continuity.
Week
10

Module 3:

6h

Unit 1: Taylor and Laurent Series

6 study hours
  • •Review Taylor and Laurent series expansions.
  • •Work through practice problems on finding series expansions.
Week
11

Module 4:

7h

Unit 1: Bilinear Transformation

7 study hours
  • •Review bilinear transformations and their applications.
  • •Work through practice problems on bilinear transformations.
Week
12

Module 1:

16h

Unit 1: Complex Numbers

8 study hours
  • •Solve additional problems on all topics covered in the course.
  • •Focus on areas where you need more practice.

Unit 2: Complex Functions

8 study hours
  • •Solve additional problems on all topics covered in the course.
  • •Focus on areas where you need more practice.
Week
13

Module 2:

16h

Unit 1: Analytic Functions

8 study hours
  • •Solve additional problems on all topics covered in the course.
  • •Focus on areas where you need more practice.

Unit 2: Limit and Continuity

8 study hours
  • •Solve additional problems on all topics covered in the course.
  • •Focus on areas where you need more practice.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Review all definitions and theorems related to complex numbers, functions, and analyticity.

2

Practice solving problems involving complex arithmetic, transformations, and limits.

3

Focus on understanding the Cauchy-Riemann equations and their applications.

4

Master the techniques for finding Taylor and Laurent series expansions.

5

Practice applying bilinear transformations to solve problems.

6

Review all tutor-marked assignments and their solutions.

7

Create concept maps linking different topics in complex analysis.

8

Allocate sufficient time for practice problems and review before the exam.

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