Real and complex analysis(MC222)

Instructor: Sudip Bera

Winter- 2024

DA-IICT, Gandhinagar

This is a core course for second year students of BTech Mathematics and computing program. It is 3 hours lecture every week and 1 tutorial per week. The primary aim of this course is to introduce students to the fundamentals of Real and complex analysis.

Course Goals: After successful completion of the course, students will be able to apply the knowledge of mathematics, science, engineering fundamentals to the solution of complex engineering problems. They can understand and create mathematical arguments for solving problems. Develop skills towards mathematical modeling and analysis of engineering problems .

Suggested Textbook/references :

• Topology of metric spaces, S. Kumaresan.

• A basic course in real analysis, S. Kumaresan and A. Kumar.

• A first course in complex analysis with applications, D. G. Zill and P. D. Shanahan.

• Complex analysis, T. Gamelin.

• Complex Analysis, L. V. Ahlfors.

Assessment method: Quizes (surprise tests)(10%), Two in-semester examinations(30% each), final examination(30%).

This course covers the following topics: Countable and uncountable sets, Definition and Examples of Metric spaces, Open Balls and Open Sets. Convergence: Convergent Sequences, Limit and Cluster Points, Cauchy Sequences and Completeness, Bounded Sets, Dense Sets, Basis, Boundary of a Set, Series and convergence criteria for series of positive real numbers , Continuous Functions, Equivalent Definitions of Continuity, Topological Property, Uniform Continuity, Limit of a Function , Compactness, Connectedness.

Complex Numbers and the Complex Plane , Complex Functions and Mappings, Analytic Functions, Integration in the Complex Plane, Series and Residues, Conformal Mappings.