Singapore University of Social Sciences

Fundamentals of Complex Analysis (MTH301)

Synopsis

Complex numbers, complex functions and the analysis of functions of one complex varaible are introduced. The understanding of the basic theory and the techniques of complex analysis is the ket to succesful applications. The course aims to introduce the basic fundamental concepts and show how they are woven together to provide a powerful tool for application.

Level: 3
Credit Units: 5
Presentation Pattern: Every January

Topics

• Complex number field.
• Complex functions.
• Sequences.
• Subsets of the complex plane.
• Continuous functions.
• Limits.
• Complex differentiation.
• Paths, rectifiable path, smooth path, length of a path.
• Contour and contour integration.
• Cauchy’s integral theorem.
• Cauchy’s integral formula.
• Taylor and Laurent series, and residues.

Learning Outcome

• Calculate contour integrals, Laurent series or Taylor series of functions of one complex variable.
• Determine the points of functions of one complex variable which are continuous/differentiable/analytic.
• Show how to prove a mathematical statement in complex analysis.
• Indicate the nature of singularities of complex functions.
• Compute limits and/or residues of functions of one complex variable.
• Demonstrate mathematical reasoning through proving mathematical statements in complex analysis.