Campus

SNS

Programs

PG

Session

Fall Semester 2016

Course Title

Analysis

Course Code

MATH802

Credit Hours

30

PreRequisutes


Course Objectives

This course is designed to provide indepth
knowledge of Real and Complex Analysis

Detail Content

Real Analysis
 The sandwiching theorem, the nested intervals theorem, the BolzanoWierstrass theorem.
 Uniform continuity, Cauchy criterion for convergence.
 The HeineBorel Theorem.
 Review of Rolle’s, meanvalue theorems and L’Hopital rule.
 Inverse functions in R1, Inverse function theorem, inverse
differentiation theorem.
 The Darboux integral for functions in R1, Upper and lower Darboux integrals, Fundamental theorems of calculus – 1st and 2nd forms, The Riemann integral.
 Continuity in RN, Darboux integral in RN, The Riemann integral in RN.
 Sequences of functions and uniform convergence of sequences.
 Uniform convergence of series
 Fourier series; Fourier sine and cosine series; Halfrange series; Convergence theorems.
 The derivative of a function defined by an integral; Leibnitz rule; Convergence and divergence of improper integrals; The Gamma function.
 Functions of bounded variation; the RiemannStieltjes integral.
 The implicit function theorems; Change of variables in multiple integrals; The Lagrange multiplier rule
Complex Analysis
 Analyticity of complex functions.
 Transcendental functions in the complex plane
 Complex integration.
 Cauchy’s integral theorem and Cauchy’s integral formula.
 Sequence and series and their convergence.
 Power series, Taylor series, and Laurent series.
 Singularities and zeros.
 Residue integration
 Evaluation of real integrals
 Conformal mapping

Text/Ref Books

 M. H. Protter, and C. B. Morrey, Jr.: A First Course in Real Analysis, Springer (1991)
 E. B. Saff, and A. D. Snider: Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Prentice Hall (1993)

Time Schedule

Fall Semester 2014

Faculty/Resource Person

Dr. Matloob Anwar
PhD GCU Lahore
Discipline: Mathematical Analysis

