Mathematics (Paper-II ) Numerical Analysis & Operations Research Book B.Sc 6th Sem U.P

Unit

Topics

No. of Lectures

Part A: Metric Spaces

I

Basic Concepts

Metric Spaces: Definition and examples, Sequences in metric spaces, Cauchy sequences, Complete metric space.

08

II

Topology of Metric Spaces

Open and closed ball, Neighborhood, Open set, Interior of a set, limit point of a set, derived set, closed set, closure of a set, diameter of a set, Cantor’s theorem, Subspaces, Dense set.

08

III

Continuity & Uniform Continuity in Metric Spaces

Continuous mappings, Sequential criterion and other characterizations of continuity, Uniform continuity, Homeomorphism, Contraction mapping, Banach fixed point theorem.

07

IV

Connectedness and Compactness

Connectedness, Connected subsets of, Connectedness and continuous mappings, Compactness, Compactness and boundedness, Continuous functions on compact spaces.

07

Part B: Complex Analysis

V

Analytic Functions and Cauchy-Riemann Equations

Functions of complex variable, Mappings; Mappings by the exponential function, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives, Differentiation formulae, Cauchy-Riemann equations, Sufficient conditions for differentiability; Analytic functions and their examples.

08

VI

Elementary Functions and Integrals

Exponential function, Logarithmic function, Branches and derivatives of logarithms, Trigonometric function, Derivatives of functions, Definite integrals of functions, Contours, Contour integrals and its examples, Upper bounds for moduli of contour integrals.

08

VII

Cauchy’s Theorems and Fundamental Theorem of Algebra

Antiderivatives, Proof of antiderivative theorem, Cauchy-Goursat theorem, Cauchy integral formula; An extension of Cauchy integral formula, Consequences of Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra.

07

VIII

Series and Residues

Convergence of sequences and series, Taylor series and its examples; Laurent series and its examples, Absolute and uniform convergence of power series, Uniqueness of series representations of power series, Isolated singular points, Residues, Cauchy’s residue theorem, residue at infinity; Types of isolated singular points, Residues at poles and its examples.

07