MA547: Complex Analysis, January - April 2025

Last updated on April 21, 2025

Understanding the geometry of complex functions presents a distinctive challenge when contrasted with the more familiar Euclidean plane. This complexity is primarily rooted in the algebraic structure of the complex plane, which is significantly shaped by its specialized multiplication properties. A pivotal element of this structure is captured by the Cauchy-Riemann equations, which provide a framework different from that of the usual Frechet derivative.

To effectively navigate the complexities inherent in complex geometry, it is essential to investigate mappings, transformations, and function representations. Fundamental concepts such as holomorphic functions, power series expansion, Cauchy theorem, maximum principle, analytic continuation, and conformal mappings form the cornerstone of complex analysis. These principles not only facilitate a deeper understanding of complex functions but also illuminate their geometric interpretations and applications across diverse fields of mathematics and engineering.

Course policy (click here)

Classroom and slot: 1G1, Slot-A (Mon, Tue, Wed, Thu)

Syllabus: Review of complex numbers; Analytic functions, harmonic functions, elementary functions, branches of multiple-valued functions, conformal mappings; Complex integration, Cauchy's integral theorem, Cauchy's integral formula, theorems of Morera and Liouville, maximum-modulus theorem; Power series, Taylor's theorem and analytic continuation, zeros of analytic functions, open mapping theorem; Singularities, Laurent's theorem, Casorati-Weierstrass theorem, argument principle, Rouche's theorem, residue theorem and its applications in evaluating real integrals.

 Texts/references:

1. R.V. Churchill and J.W. Brown, Complex Variables and Applications, 5th edition, McGraw Hill, 1990.
2. J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 3rd edition, Narosa, 1998.
3. L. V. Ahlfors, Complex Analysis, 3rd Edn., McGraw Hill, 1979.
4. J. E. Marsden and M. J. Hoffman, Basic complex analysis, 3rd Edn., W. H. Freeman, 1999.
5. D. Sarason, Complex function theory, 2nd Edn., Hindustan book agency, 2007.
6. J.B. Conway, Functions of One Complex Variable, 2nd Edn., Narosa, 1973.
7. Bruce P. Palka, An Introduction to Complex Function Theory (1995, Springer)
8. Reinhold Remmert, Theory of complex functions, 122, GTM

 Some helpful references:

I. Complex Analysis by T. Gamelin

II. A Collection of Problems on Complex Analysis, by I.G. Aramanovich, L G.L. Lunts, and .I. Volkovyskii  
[It will help understand the applications of complex analysis as well.]

III. Complex Analysis with Applications, by Loukas Grafakos and Nakhle H. Asmar

IV. What Is Mathematics? by R. Courant, H. Robbins, and I. Stewart

V. Our Mathematical Universe, by M. Tegmark

VI. Don't Believe Everything You Think, by J. Nguyen [ For more similar references (click here)]

A useful course link:

The MA201: Mathematics III, 2024 slides could be an unfolding tool for understanding early bird basic ideas of complex analysis (click here)

Lecture notes: Lecturenotes 1, Lecturenotes 2, Lecturenotes 3, Lecturenotes 4, Lecturenotes 5

Assignments: Assignment 1, Assignment 2, Assignment 3, Assignment 4, Assignment 5

Exams: Quiz-I, Mid-Sem, Quiz-II

 Class Discipline:

 Please ensure your mobile phone is off or on silent and kept in your bag under the desk during class. Only a notebook and pen should be on the desk. Any student with a phone on their desk will be asked to leave and reported to the academic section.