Metric Spaces and Complex Analysis

Overview

Learners having idea of fundamental Mathematics can easily understand the fundamentals of functions of a complex variable, metric spaces, and various theorems like Cantor’s theorem, Banach Fixed point Theorem, Cauchy-Riemann equations, Cauchy-Goursat theorem, Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.The main objectives areTo understand the concept of a metric space, to familiarize the ideas of open and closed sets, to learn the concept of continuity, homeomorphism and connectedness, to provide a foundation for more advanced courses in Mathematical analysis, to provide a new perspective on many of the ideas studied in Real Analysis, to study the techniques of complex variables and functions together with their derivatives, Contour integration and transformations and developing a clear understanding of the fundamental concepts of Complex Analysis

Syllabus

Weeks Weekly Lecture Topics (Module Titles) 1 Day 1 Module 1 : METRIC SPACES AND EXAMPLES Day 2Module 2 : SEQUENCES IN METRIC SPACES Day 3Module 3 : OPEN SETS Day 4 Day 5 2 Day 1 Module 4 : FUNDAMENTAL PROPERTIES OF OPEN SETS Day 2Module 5 : CLOSED SETS Day 3Module 6 :CANTOR SET AND CLOSURE OF A SET Day 4 Day 5 3 Day 1 Module 7 : BOUNDARY OF A SET AND DENSE SET Day 2Module 8 : THEOREMS ON OPEN AND CLOSED SETS Day 3Module 9 : SEPARABLE SPACES Day 4 Day 5 4 Day 1 Module 10 : CONTINUITY Day 2Module 11 : UNIFORM CONTINUITY Day 3Module 12 : BAIRE'S THEOREM Day 4 Day 5 5 Day 1 Module 13 : HOMEOMORPHISM Day 2Module 14 : CONNECTEDNESS Day 3Module 15 : PROPERTIES OF COMPLEX NUMBERS Day 4 Day 5 6 Day 1 Module 16 : POLAR AND EXPONETIAL FORM Day 2Module 17 : FUNCTIONS OF A COMPLEX VARIABLE Day 3Module 18 : LIMIT OF FUNCTIONS OF A COMPLEX VARIABLE Day 4 Day 5 7 Day 1 Module 19 : POINT AT INFINITY Day 2Module 20 : CONTINUITY OF FUNCTIONS OF A COMPLEX VARIABLE Day 3Module 21 : MAPPINGS Day 4 Day 5 8 Day 1 Module 22 : DIFFERENTIATION OF FUNCTIONS OF A COMPLEX VARIABLE Day 2Module 23: CAUCHY-RIEMANN EQUATIONS - I Day 3Module 24 : CAUCHY-RIEMANN EQUATIONS - II Day 4 Day 5

9 Day 1 Module 25 : ANALYTIC FUNCTIONS Day 2Module 26 : EXPONENTIAL FUNCTIONS Day 3Module 27: TRIGONOMETRIC FUNCTIONS Day 4 Day 5 10 Day 1 Module 28 : LOGARITHMIC FUNCTIONS Day 2Module 29 : HARMONIC FUNCTIONS Day 3Module 30 : DEFINITE INTEGRALS Day 4 Day 5 11 Day 1 Module 31 : CONTOURS Day 2Module 32 : CONTOUR INTEGRALS Day 3Module 33: CAUCHY-GOURSAT THEOREM Day 4 Day 5 12 Day 1 Module 34 : CAUCHY'S INTEGRAL FORMULA Day 2Module 35 : LIOVILLE'S THEOREM Day 3Module 36 : SEQUENCES AND SERIES - I Day 4 Day 5 13 Day 1 Module 37 :SEQUENCES AND SERIES - II Day 2Module 38 : TAYLOR SERIES Day 3Module 39 : LAURENT SERIES Day 4 . Day 5 14 Day 1 Module 40 : POWER SERIES Day 2 Day 3 Day 4