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Introduction to Real Analysis (FOR DEGREE HONOURS COURSE) by Sadhan Kumar Mapa


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Introduction to Real Analysis (FOR DEGREE HONOURS COURSE) written by Sadhan Kumar MapaReader in Mathematics (retired), Presidency College, Calcutta . This is an other book of mathematics cover the following topics.

  • l. Set Theory
    1.1. Introduction
    1 1.2. Subsets
    1 1.3. Algebraic operations on sets
    2 1.4. Family of sets
    1.5. Cartesian product of sets
    4 5 1.6. Relation on a set . ~
    5 1.7. Order relation on a set
    6 1.8. Function
    1.9. Equipotent sets, enumerable set
    10 Exercises 1

  • 2. Real numbers
    2.l. Natural numbers
    2.2. Integers
    2.3. Rational numbers
    2.4. Real numbers
    Exercises 2

  • 3. Sets in R
    3.1. Intervals
    3.2. Neighbourhood
    3.3. Interior point
    3.4. Open set
    3.5. Limit point
    3.6. Isolated point
    3.7. Derived set
    3.8. Closed set
    3.9. Adherent point
    3.10. Dense set, Perfect set
    Exercises 3
    3.11. Nested intervals
    3.12. Decimal representation
    3.13. Enumerable set
    Exercises 4
    3.14. Point of condensation
    3,15. Borel set 85
    3.16. Cover,. open cover
    Exercises 5

  • 4. Real functions
    4.1. Real function
    4.2. Injective function, Surjective function
    4.3. Equal functions
    4.4. Restriction function
    4.5. Composite function
    4.6. Inverse function
    4.7. Algebraic operations on functions
    4.8. Monotone functions
    4.9. Even function; odd function
    4.10. Power function
    4.11. Exponential fUnction
    4.12. Logarithmic function
    4.13. Hyperbolic functions
    4.14. Bounded function
    Exercises 6

  • 5. Sequence
    5.1. Real sequence
    5.2. Bounded sequence
    5.3. Limit of a sequence
    5.4. Convergent sequence
    5.5. Limit theorems
    5.6. Null sequence
    5.7. Divergent sequence
    5.8. Some important limits
    5.9. Monotone sequence
    5.10. Some important sequences
    Exercises 7
    5.11. Subsequence
    5.12. Subsequential limit
    5.13. Characterisation of a compact set
    5.14. Upper limit and lower limit
    5.15. Cauchy criterion
    5.16. Cauchy's theorems on limits
    Exercises 8

  • 6. Series
    6.1. Infinite series
    6.2. Series of positive terms
    6.3. Tests for convergence
    Exercises 9
    6.4. Series of arbitrary terms
    6.5. Conditionally convergent series
    6.6. Multiplication of series
    Exercises 10

  • 7. Limits
    7.1. Limit of a function
    7.2. One-sided limits
    7.3. Infinite limits
    7.4. Limits at infinity
    7.5. Infinite limits at infinity
    7.6. Limits of monotone functions
    7.7. Some important'limits
    Exercises 11

  • 8. Continuity
    8.1. Continuity
    8.2. Continuity of some important functions
    8.3. Limit of composite functions
    8.4. Discontinuity
    Exercises 12
    8.5. Properties of continuous functions
    8.6. Monotone functions and continuity
    8.7. Uniform continuity
    8.8. Continuity on a compact set
    Exercises 13

  • 9. Differentiation
    9.1. Differentiability. Derivative
    9.2. Higher order derivatives
    Exercises 14
    9.3. Sign of the derivative
    9.4. Properties of the derivative
    9.5. Rolle's theorem and Mean value theorems
    Exercises 15
    9.6. The nth order derivatives
    Exercises 16
    9.7. Taylor's theorem and expansion of functions
    Exercises 17
    9.8. Maxima and minima
    Exercises 18
    9.9. Indeterminate forms
    Exercises 19

  • 10. Functions of bounded variation
    10.1. Introduction
    10.2. Variation function
    10.3. Positive variation, negative variation
    Exercises 20

  • 11. Riemann integral
    11.1. Partition
    11.2. Riemann integrability
    11.3. Refinement of a partition
    11.4. Norm of a partition
    11.5. Some Riemann integrable functions
    11.6. Properties of Riemann integrable functions
    11.7. Inequalities
    11.8. Fundamental theorem
    11:9. Another definition of integrability
    11.10. Integration by substitution
    11.11. Integration by parts
    11.12. Mean value theorems
    11.13. Logarithmic function
    11.14. Exponential function
    Exercises 21

  • 12. Improper integrals
    12.1. Introduction
    12.2. Definitions
    12.3. Tests for convergence (positive integrand)
    12.4. Tests for convergence
    12.5. Definitions
    12.6. Tests for convergence (positive integrand)
    12.7. Tests for convergence
    12.8. Tests for convergence of the integral of a product
    12.9. Some theorems
    12.10. Evaluation of some improper integrals
    Exercises 22
    12.11. Beta function and Gamma function ~
    Exercises 23

  • 13. Sequence of functions
    13.1. Sequence of functions
    13.2. Pointwise convergence
    13.3. Uniform convergence
    13.4. Consequences of uniform convergence

  • 14 Series of functions
    14.1. Uniform convergence
    14.2. Consequences of uniform convergence
    14.3. Abel's and Dirichlet's tests
    Exercises 25

  • 15. Power series J .
    15.1. Introduction
    15.2. Determination of radius of convergence
    15.3. Properties of a power series
    Exercises 26

  • APPENDIX 1- Sets in IR?

  • APPENDIX 2- Sequence in IR2

  • ANSWERS TO EXERCISES

  • BIBLIOGRAPHY

  • INDEX

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