Complex Analysis I
- Sciences
- 300 level
- 3 credit units
- 71 pages
- 6 units
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.
About this course
- Difficulty
- Intermediate
- Study hours
- 150 hours
- Maths
- Advanced
- Content
- Theoretical, problem solving
- Practical work
- No
- Calculus I
- Calculus II
- Assignments
- Tutor marked assignments
- End of course examination
What you'll read
The real module and unit structure of MTH304, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH304 · UNIT 1 COMPLEX NUMBERS
The objectives of this course is to teach you Complex Analysis while also acquainting you with the graphical and mathematical significance of Complex numbers and functions and their applications to “Taylor and Laurent Series and Bilinear Transformation” .All of the above are expected to motivate you towards further enquiry into this very interesting and highly specialised mathematical habitat.
What you should be able to do
- Apply the concept of complex numbers to solve mathematical problems.
- Analyze complex functions and their properties.
- Determine whether a given function is analytic.
- Evaluate limits and determine the continuity of complex functions.
- Find Taylor and Laurent series expansions of functions.
- Apply bilinear transformations to solve problems.
What it prepares you for
- Mathematician
- Engineer
- Data Analyst
- Financial Analyst
- Software Developer
- Engineering
- Physics
- Computer Science
- Finance
- Telecommunications
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 2:
Unit 1: Analytic Functions
The Cauchy-Riemann equations require a strong understanding of partial derivatives and their relationship to complex differentiability, which can be challenging for students new to complex analysis.
- Module 3:
Unit 1: Taylor and Laurent Series
Laurent series involve both positive and negative powers, making it difficult to determine the region of convergence and the appropriate expansion for a given function.
A suggested way through it
13 weeks, about 100 hours in total. Yours will differ.
- Week 1Module 1:
Unit 1: Complex Numbers · 4 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 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 2Module 2:
Unit 1: Analytic Functions · 5 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 3Module 2:
Unit 2: Limit and Continuity · 5 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 4Module 3:
Unit 1: Taylor and Laurent Series · 6 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 5Module 3:
Unit 1: Taylor and Laurent Series · 6 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 6Module 4:
Unit 1: Bilinear Transformation · 7 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 7Module 1:
Unit 1: Complex Numbers · 4 hours
Review complex numbers, functions, and their properties.. Work through practice problems on complex arithmetic and transformations..
Unit 2: Complex Functions · 4 hours
Review complex functions and their properties.. Work through practice problems on complex functions and transformations..
- Week 8Module 2:
Unit 1: Analytic Functions · 5 hours
Review analytic functions and the Cauchy-Riemann equations.. Work through practice problems on determining analyticity..
- Week 9Module 2:
Unit 2: Limit and Continuity · 5 hours
Review limits and continuity of complex functions.. Work through practice problems on limits and continuity..
- Week 10Module 3:
Unit 1: Taylor and Laurent Series · 6 hours
Review Taylor and Laurent series expansions.. Work through practice problems on finding series expansions..
- Week 11Module 4:
Unit 1: Bilinear Transformation · 7 hours
Review bilinear transformations and their applications.. Work through practice problems on bilinear transformations..
- Week 12Module 1:
Unit 1: Complex Numbers · 8 hours
Solve additional problems on all topics covered in the course.. Focus on areas where you need more practice..
Unit 2: Complex Functions · 8 hours
Solve additional problems on all topics covered in the course.. Focus on areas where you need more practice..
- Week 13Module 2:
Unit 1: Analytic Functions · 8 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 hours
Solve additional problems on all topics covered in the course.. Focus on areas where you need more practice..
Preparing for the exam
- Review all definitions and theorems related to complex numbers, functions, and analyticity.
- Practice solving problems involving complex arithmetic, transformations, and limits.
- Focus on understanding the Cauchy-Riemann equations and their applications.
- Master the techniques for finding Taylor and Laurent series expansions.
- Practice applying bilinear transformations to solve problems.
- Review all tutor-marked assignments and their solutions.
- Create concept maps linking different topics in complex analysis.
- Allocate sufficient time for practice problems and review before the exam.
Questions students ask about this course
What is MTH304 about?
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.
How many units does MTH304 have?
MTH304, Complex Analysis I, has 6 units across 4 modules, over 71 pages of course material. You can read it one unit at a time.
How many credit units is MTH304?
MTH304 carries 3 credit units, at 300 level in Sciences.
Is MTH304 hard?
MTH304 is rated intermediate level, with advanced mathematical content. It is mostly theoretical and problem solving work.
How long does MTH304 take to study?
About 150 hours of study, spread across its 6 units.
How is MTH304 assessed?
MTH304 is assessed by assignments, tutor marked assignments and end of course examination.
What do I need before starting MTH304?
Calculus I Calculus II
What can I do with MTH304?
Mathematician, Engineer, Data Analyst, Financial Analyst and Software Developer.