Syllabus (MATHEMATICS)

Course Type: MAJ-11

Semester: 7

Course Code: BMTMMAJ11T

Course Title: Metric Space & Complex Analysis

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: CO1: The course is aimed at exposing the students to foundations of analysis, which will be useful in understanding various physical phenomena and give the students the foundation in

Syllabus:

Unit-1: Metric Spaces [Credit: 3, 45L]

• Metric, examples of standard metric spaces including Euclidean and Discrete metrics; open ball, closed ball, open sets; metric topology; closed sets, limit points and their fundamental properties; interior, closure and boundary of subsets and their interrelation; denseness; separable and second countable metric spaces and their relationship.
• Continuity: Definition of continuous functions, algebra of real/complex valued continuous functions, distance between a point and a subset, distance between two subsets, Homeomorphism (definitions with simple examples)
• Sequence and completeness: Sequence, subsequence and their convergence; Cauchy sequence, Cauchy’s General Principle of convergence, Cauchy’s Limit Theorems. Completeness, completeness of R; Cantor’s theorem concerning completeness, Definition of completion of a metric space, construction of the real as the completion of the incomplete metric space of the rational with usual distance (proof not required). Continuity preserves convergence.
• Compactness: Sequential compactness, Heine-Borel theorem in R. Finite intersection property, continuous functions on compact sets.
• Connectedness: Connected subsets of the real line R, open connected subsets in R2, components; components of open sets in R and R2; Structure of open set in R, continuity and connectedness; Intermediate value theorem.

Unit-2: Complex Analysis [Credit: 2, 30L]

• Introduction of complex number as ordered pair of real numbers, geometric interpretation, metric structure of the complex plane C, regions in C. Stereographic projection and extended complex plane C_∞ and circles in C_∞ .
• Limits, Continuity and differentiability of a function of complex variable, sufficient condition for differentiability of a complex function, Analytic functions and Cauchy-Riemann equation, harmonic functions, Conjugate harmonic functions, Relation between analytic function and harmonic function.
• Transformation (mapping), Concept of Conformal mapping, Bilinear or Mobius transformation and its geometrical meaning, fixed points and circle preserving character of Mobius transformation.
• Power series: Cauchy-Hadamard theorem. Determination of radius of convergence. Uniform and absolute convergence of power series. Analytic functions represented by power series. Uniqueness of power series.
• Contours, complex integration along a contour and its examples, upper bounds for moduli of contour integrals. Cauchy-Goursat theorem (statement only) and its consequences, Cauchy integral formula.

Reading References:

1. Satish Shirali and Harikishan L. Vasudeva; Metric Spaces; Springer Verlag, London; 2006.
2. S. Kumaresan; Topology of Metric Spaces, 2nd Ed.; Narosa Publishing House; 2011.
3. J. Sengupta; Metric Spaces; U.N. Dhur & Sons Private Limited.
4. P. K. Jain and K. Ahmad; Metric Spaces; Narosa Publishing House.
5. G.F. Simmons; Introduction to Topology and Modern Analysis; McGraw-Hill; 2004.
6. James Ward Brown and Ruel V. Churchill; Complex Variables and Applications, 8th Ed.; McGraw – Hill International Edition; 2009.
7. J.B. Conway; Functions of one Complex Variable, 2nd Ed.; Undergraduate Texts in Mathematics; Springer-Verlag New York, Inc.
8. S. Ponnusamy; Foundations of complex analysis; Narosa Publishing House
9. E.M. Stein and R. Shakrachi; Complex Analysis, 2nd Ed; Princeton University Press.
10. Joseph Bak and Donald J. Newman; Complex Analysis, 2nd Ed.; Undergraduate Texts in Mathematics; Springer-Verlag New York, Inc., New York; 1997.
11. R. Roopkumar; Complex Analysis; Pearson.