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MTH305Sciences3 Unitsintermediate

Complex Analysis Ii

This course delves into the analysis of complex variables, establishing results analogous to real number systems. It covers functions of complex variables, their limits, continuity, and convergence of sequences and series. Key topics include transformations, elementary functions, Taylor and Laurent series, and the Cauchy-Riemann equations. The course also explores singularities, residues, and complex integration, equipping students with essential tools for advanced mathematical analysis.

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156h
Study Time
13
Weeks
12h
Per Week
advanced
Math Level
Course Keywords
Complex VariablesAnalytic FunctionsCauchy-Riemann EquationsLaurent SeriesComplex Integration

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

Analytic Functions

3

Cauchy-Riemann Equations

4

Taylor Series

5

Laurent Series

6

Complex Integration

7

Residue Theorem

8

Harmonic Functions

Total Topics8 topics

Ready to Start

No specific requirements needed

This course is designed to be accessible to all students. You can start immediately without any prior knowledge or specific preparation.

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

final examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

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Aerospace Engineer

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Electrical Engineer

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Applied Mathematician

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Data Scientist

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Financial Analyst

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Industry Applications

Real-world sectors where you can apply your knowledge

AerospaceTelecommunicationsFinanceData AnalysisCryptography

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Introduction

4h

Unit 1: Function of Complex Variables

4 study hours
  • •Understand the definition of complex variables and functions.
  • •Study transformations and their properties.
  • •Familiarize yourself with elementary functions like polynomials and exponentials.
Week
2

Module 1: Introduction

4h

Unit 2: Limits and Continuity of Function of Complex Variables

4 study hours
  • •Learn the definitions of limits and continuity for complex functions.
  • •Study theorems related to limits and continuity.
  • •Practice solving problems involving limits and continuity.
Week
3

Module 1: Introduction

4h

Unit 3: Convergence of Sequence and Series of Complex Variables

4 study hours
  • •Understand the convergence of sequences and series of complex variables.
  • •Study related theorems for complex variables.
  • •Solve problems on series and sequences.
Week
4

Module 1: Introduction

4h

Unit 4: Some Important Theorems

4 study hours
  • •Review and understand important theorems related to complex variables.
  • •Focus on theorems related to convergence.
  • •Practice applying these theorems to solve problems.
Week
5

Module 2: Advanced Topics

4h

Unit 1: Some Examples on Taylor and Laurent Series

4 study hours
  • •Study examples of Taylor and Laurent series expansions.
  • •Practice expanding functions using Taylor and Laurent series.
  • •Determine the regions of convergence for these series.
Week
6

Module 2: Advanced Topics

4h

Unit 2: Analytic Functions

4 study hours
  • •Learn about derivatives of complex variables.
  • •Understand the Cauchy-Riemann equations.
  • •Study harmonic functions and their properties.
Week
7

Module 2: Advanced Topics

4h

Unit 3: Principles of Analytic Continuation

4 study hours
  • •Examine the principles of analytic continuation.
  • •Understand residues and the residue theorem.
  • •Learn to calculate residues.
Week
8

Module 2: Advanced Topics

4h

Unit 4: Complex Integration

4 study hours
  • •Study curves, simply and multiply connected regions.
  • •Understand complex line integrals.
  • •Learn the Cauchy-Goursat theorem.
Week
9

Module 1: Introduction

8h

Unit 1: Function of Complex Variables

4 study hours
  • •Review Module 1 Units 1-2
  • •Practice problems related to functions, limits, and continuity of complex variables.
  • •Solve additional exercises to reinforce understanding.

Unit 3: Convergence of Sequence and Series of Complex Variables

4 study hours
  • •Review Module 1 Units 3-4
  • •Practice problems related to convergence, sequences, and important theorems.
  • •Solve additional exercises to reinforce understanding.
Week
10

Module 2: Advanced Topics

8h

Unit 1: Some Examples on Taylor and Laurent Series

4 study hours
  • •Review Module 2 Units 1-2
  • •Practice problems related to Taylor and Laurent series, and analytic functions.
  • •Solve additional exercises to reinforce understanding.

Unit 3: Principles of Analytic Continuation

4 study hours
  • •Review Module 2 Units 3-4
  • •Practice problems related to analytic continuation and complex integration.
  • •Solve additional exercises to reinforce understanding.
Week
11

Module 1: Introduction

6h

Unit 1: Function of Complex Variables

6 study hours
  • •Work on Tutor-Marked Assignment for Module 1
  • •Focus on applying concepts from Units 1-4.
  • •Ensure all questions are answered thoroughly and accurately.
Week
12

Module 2: Advanced Topics

6h

Unit 1: Some Examples on Taylor and Laurent Series

6 study hours
  • •Work on Tutor-Marked Assignment for Module 2
  • •Focus on applying concepts from Units 1-4.
  • •Ensure all questions are answered thoroughly and accurately.
Week
13

Module 1: Introduction

8h

Unit 4: Some Important Theorems

8 study hours
  • •Final Revision
  • •Review all course materials, focusing on key concepts and theorems.
  • •Practice solving a variety of problems to prepare for the final examination.

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

Thoroughly review all definitions and theorems related to complex functions and their properties.

2

Practice expanding functions using Taylor and Laurent series, paying close attention to regions of convergence.

3

Master the application of the Cauchy integral formula and residue theorem for evaluating complex integrals.

4

Focus on understanding and applying the Cauchy-Riemann equations to determine the analyticity of complex functions.

5

Solve a variety of problems from each unit, including those from the tutor-marked assignments, to reinforce understanding.

6

Create concept maps linking key concepts from different modules to see the connections between them.

7

Allocate specific time slots each week for focused study and revision of complex analysis topics.

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