Course Title: COMPLEX ANALYSIS
Level: Graduate Course
Type: Elective
No. of Credits: 4
Pre-requisites: None
Course Rationale
Complex analysis is the study of functions of complex variables. It is a diverse subject, with connections to numerous parts of mathematics, such as the prime number theorem, the fundamental theorem of algebra, the Jordan canonical form theorem for matrices, the Weierstrass approximation theorem, the evaluation of "improper" integrals, the geometry of transformations, and the harmonic functions of fluid mechanics.
This course will therefore benefit not only the students enrolled in the Master of Science in Mathematics program, but also the students throughout the sciences and engineering who require fluency with the manipulations and results of complex analysis.
Course Description
The course develops the properties of the complex number system, treated as a generalization of the real number system. We explore the parallel analysis that results, with a particular emphasis on differentiability, analyticity, contour integrals, Cauchy’s theorem, Laurent series representation, and residue calculus.
Core topics include: complex numbers, analytic functions and their properties, derivatives, integrals, series representations, residues, and conformal mappings. Application of the calculus of residues and mapping techniques to the solution of common boundary value problems encountered in physics and engineering applications is a major part of the course.
Students are expected to have a strong background in advanced undergraduate calculus of real variables. An earlier or concurrent course in differential equations is an asset, but is not a prerequisite for this course.
Learning Outcomes
Upon successful completion of the course, students will be able to:
- Solve inequalities involving complex numbers
- Utilize the spherical representation of complex numbers
- Distinguish between analytic and non-analytic functions
- State the necessary and sufficient conditions for a function to be analytic
- Outline the Heine-Borel property and apply the Bolzano-Weierstrass theorem
- Create and describe conformal mappings
- Evaluate line integrals
- State and use the Cauchy Theorems to evaluate complex integrals
- Identify and classify the singularities of complex functions
- Calculate the residue of functions at poles
- Predict the behavior of a function in the neighbourhood of a singularity via Weierstrass’ Theorem
- Calculate complex integrals via the Residue Theorem
- Apply relevant theorems to evaluate definite integrals in the complex plane
- Determine the power series, Taylor series and Laurent series representations for complex functions
Content (48 hours)
- Introduction: Inequalities involving complex numbers. The spherical representation of the complex plane (2 hours)
- Limits and Continuity: Analytic Functions, Necessary and Sufficient conditions (Cauchy-Riemann equations). Polynomials, Lucas’s theorem, Rational Functions (5 hours)
- Connectedness, compactness in C, Heine-Borel Property in C, Bolzano-Weierstrass theorem for C, Continuous Functions on compact subsets of C, Arcs and closed curves, Analytic Functions in regions, Conformal mapping, Linear transformation, The linear group, The cross ratio, Symmetry. (12 hours)
- Complex Integration: Line integrals, Rectifiable arcs, Line integrals as functions of arcs, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disk, The index of a point with respect to a closed curve. Cauchy’s integral formula. Higher derivatives. Morera’s theorem, Liouville’s theorem (12 hours)
- Singularities and Residues :Classical theorem of Weierstrass concerning behaviour of a function in the neighbourhood of an essential singularity, The Jocal mapping, open mapping theorem The maximum principle, Schwarz’s lemma, The residue theorem, The argument principle, Rouche’s theorem, Evaluation of definite integrals.(12 hours)
- Series Expansions: Power series expansions, The Weierstrass theorem, The Taylor series, The Laurent series. (5 hours)
Teaching Methodology
Lectures: Three (3) lectures each week (50 minutes each).
Tutorials: One (1) weekly tutorial session (50 minutes of problem solving, based on theory covered during lectures).
Assessment
Coursework Mark: 40% - Two 15% Coursework examinations and 10% Assignments (*based on four assignments given during the semester)
Final Examination: 60% (one 3-hour written paper)
Course Calendar
Week |
Lecture/Tutorial Topic |
Assignments |
---|---|---|
1 |
Introduction: Inequalities involving complex numbers. The spherical representation of the complex plane. Limits and Continuity: Analytic Functions |
- |
2 |
Limits and Continuity: Analytic Functions: Necessary and Sufficient conditions (Cauchy-Riemann equations). Connectedness, compactness in C: Heine-Borel Property in C, |
Assignment #1 |
3 |
Connectedness, compactness in C: Bolzano-Weierstrass theorem for C, Continuous Functions on compact subsets of C, Arcs and closed curves. |
- |
4 |
Connectedness, compactness in C: Analytic Functions in regions, Conformal mapping, Linear transformation. |
Assignment #2 |
5 |
Connectedness, compactness in C: The linear group, The cross ratio, Symmetry Complex Integration: Line integrals. |
- |
6 |
Complex Integration: Rectifiable arcs, Line integrals as functions of arcs, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disk |
Coursework Exam #1 |
7 |
Complex Integration: The index of a point with respect to a closed curve. Cauchy’s integral formula. |
- |
8 |
Complex Integration: Higher derivatives. Morera’s theorem, Liouville’s theorem |
Assignment #3 |
9 |
Singularities and Residues :Classical theorem of Weierstrass concerning behaviour of a function in the neighbourhood of an essential singularity |
- |
10 |
Singularities and Residues: The Jocal mapping, open mapping theorem The maximum principle, Schwarz’s lemma. |
Coursework Exam #2 |
11 |
Singularities and Residues: The residue theorem, The argument principle, Rouche’s theorem, Evaluation of definite integrals. Series Expansions: Power series expansions |
Assignment #4 |
12 |
Series Expansions: The Weierstrass theorem, The Taylor series, The Laurent series |
- |
13 |
Revision |
- |
Suggested Texts / References
Texts Books:
- J. B. Conway, Functions of One Complex Variable, Springer-verlag international student edition 2. S. Lang, complex Analysis, Addison-Wesley, 1977.
- L. V. Ahlfors, Complex Analysis, Mc-Graw-Hill, 1979.
Reference Books:
- S. Ponnusamy, Foundations of Complex Analysis (2nd Edition), Alpha Science International Ltd, 2006.
- R. Remmert, Theory of Complex Functions, Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991.
- J.W. Brown and R.V. Churchill. Complex Variables and Applications (8th Edition), Mc Graw-Hill, 2008.
- S. Narayan, Theory of Functions of a Complex Variable, (P.K. Mittal – Editor) S. Chand & Co Ltd, 2005.
- E. Kreyszig, Advanced Engineering Mathematics (8th Edition), John Wiley and Sons, New York, 2001.