Syllabus For The Subject REAL ANALYSIS

UNIT - 01

DERIVATIVES: DEFINITION OF DERIVATIVES - DERIVATIVES AND CONTINUITY - ALGEBRA

OF DERIVATIVES - THE CHAIN RULE - ONE SIDED DERIVATIVES AND INFINITE

DERIVATIVES FUNCTION WITH NON ZERO DERIVATIVES AND LOCAL EXTREMA.

UNIT - 02

ROLLER THEORM - THE MEAN VALUE THEORM FOR DERIVATIVES - INTERMEDIATE -

VALUE THEORM FOR DERIVATIVES AND TAYLOR`S FORMULA WITH REMAINDER.

UNIT - 03

THE RIEMANN - STIELTIES INTEGRAL - THE DEFINITION OF RIEMANN - STIELTIES

INTEGRAL - LINEOR PROPERTIES - INTEGRATION BY PARTS - CHANGE OF VARIABLES IN

A RIEMANN - STIELTJIES INTEGRAL - REDUCTION TO A RIEMAN STEP FUNCTION AS

INTEGRATORS.

UNIT - 04

REDUCTION OF A RIEMANN - STIELTJES INTEGRAL TO A FINITE SUM - EULER`S

SUMMATION FORMLA - MONOTONICALLY INCREASING INTEGRATIONS - UPPER AND

LOWER INTEGRALS - ADDITIVE AND LINEAR PROPERTIES OF UPPER AND LOWER

INTEGRALS - RIEMAN`S CONDITION.

UNIT - 05

COMPARISON THEORMS - INTEGRATORS OF BOUNDED VARIATION - SUFFICIENT

CONDITION - EXISTENCE OF RIEMAN - STIELTJES INTEGRALS AND NECESSARY

CONDITIONS FOR EXISTENCE OF RIEMANN - STIELTJES INTEGRAL.

UNIT - 06

RIEMANN - STIELTJES INTEGRALS CONDINUED: MEON - VALUE THEORMS FOR RIEMANN

- STIELTJES INTEGRALS - THE INTEGRAL AS A FUNCTION OF THE INTERVAL - SECOND

FUNDAMENTAL THEORM OF INTEGRAL CALCULUS.

UNIT - 07

RIEMANN INTEGRAL - SECOND MEAN - VALUE THEOREM FOR RIEMANN - INTEGRALS -

REMAN - STIELTJES INTEGRALS DEPENDING ON A PARAMETER - DIFFERENTIATION

UNDER THE INTEGRAL SIGN AND INTERCHANGING THE ORDER OF INTEGRATION.

UNIT - 08

INFINITE PRODUCTS: INFINITE PRODUCTS TEST FOR CONVERGENCE OF PRODUCT -

ABSOLUTE CONVERGENCE - REARRANGEMENT OF FACTORS IN A PRODUCT -

TANNERYS THEORM - INFINITE PRODUCT FOR TRIGONOMETRIC FUNCTIONS AND

HYPER BOLIC FUNCTIONS AND BERNOULLIS NUMBERS.

UNIT - 09

LEBESQUE MEASURE: OUTER MEASURE - MEASURABLE SETS AND LEBESQUE

MEASURE - A NON MESURABLE SET - MEASURABLE FUNCTIONS AND LITTLE WOODS

THREE PRINCIPLES.

Vinayaka Missions University,Directorate of Distance Education

Salem India

MASTER OF SCIENCE IN MATHEMATICS

1 Yr.

REAL ANALYSIS{MMAT.02}(2030507)

UNIT - 10

LEBSEQUE INTEGRAL: LEBESQUE INTEGRAL OF BOUNDED MEASURABLE FUNCTION

OVER A SET OF FINITE MEASURE INTEGRAL OF A NON NEGATIVE FUNCTION.

UNIT - 11

GENERAL LEBSEQUE INTEGRAL - DERIVATIVE OF MONOTONIC FUNCTION - FUNCTIONS

OF BOUNDED VARIATION.

UNIT - 12

DERIVATION OF AND INTEGRAL - ABSOLUTE CONTINUITY.