Mathematical Analysis

Front Cover
New Age International, Jan 1, 1992 - Mathematical analysis - 903 pages
10 Reviews
The Book Is Intended To Serve As A Text In Analysis By The Honours And Post-Graduate Students Of The Various Universities. Professional Or Those Preparing For Competitive Examinations Will Also Find This Book Useful.The Book Discusses The Theory From Its Very Beginning. The Foundations Have Been Laid Very Carefully And The Treatment Is Rigorous And On Modem Lines. It Opens With A Brief Outline Of The Essential Properties Of Rational Numbers And Using Dedekinds Cut, The Properties Of Real Numbers Are Established. This Foundation Supports The Subsequent Chapters: Topological Frame Work Real Sequences And Series, Continuity Differentiation, Functions Of Several Variables, Elementary And Implicit Functions, Riemann And Riemann-Stieltjes Integrals, Lebesgue Integrals, Surface, Double And Triple Integrals Are Discussed In Detail. Uniform Convergence, Power Series, Fourier Series, Improper Integrals Have Been Presented In As Simple And Lucid Manner As Possible And Fairly Large Number Solved Examples To Illustrate Various Types Have Been Introduced.As Per Need, In The Present Set Up, A Chapter On Metric Spaces Discussing Completeness, Compactness And Connectedness Of The Spaces Has Been Added. Finally Two Appendices Discussing Beta-Gamma Functions, And Cantors Theory Of Real Numbers Add Glory To The Contents Of The Book.
 

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Very good book with proper explanation.

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math anaalysis

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Contents

REAL NUMBERS
1
OPEN SETS CLOSED SETS
33
Closure of a Set
42
Countable and Uncountable Sets
49
INFINITE SERIES
109
Positive Term Series
114
Comparison Tests for Positive Term Series
118
Cauchys Root Test
124
The Weierstrass Approximation Theorem
440
POWER SERIES
446
Properties of Functions Expressible as Power Series
450
Abels Theorem
453
FOURIER SERIES
463
Some Preliminary Theorems
465
The Main Theorem
471
Intervals Other Than n it
479

DAlemberts Ratio Test
125
Raabes Test
127
Logarithmic Test
131
Integral Test
132
Gausss Test
135
Series with Arbitrary Terms
139
Rearrangement of Terms
148
FUNCTIONS OF A SINGLE VARIABLE
154
Continuous Functions
165
Functions Continuous on Closed Intervals
174
Uniform Continuity
179
The Derivative
185
Continuous Functions
188
Increasing and Decreasing Functions
191
Darbouxs Theorem
194
Rolles Theorem
195
Lagranges Mean Value Theorem
196
Cauchys Mean Value Theorem
198
Higher Order Derivatives
206
APPLICATIONS OF TAYLORS THEOREM 216 2 Extreme Values Definitions
216
Indeterminate Forms
223
Power Series
236
Exponential Functions
238
Logarithmic Functions
240
Trigonometric Functions
243
Functional Equations
249
Functions of Bounded Variation
251
VectorValued Functions
262
THE RIEMANN INTEGRAL 270 270 1 Definitions and Existence of the Integral
270
Refinement of Partitions
277
Darbouxs Theorem
280
Conditions of Integrability
281
Integrability of the Sum and Difference of Integrable Functions
284
The Integral as a Limit of Sums Riemann Sums
293
Some Integrable Functions
300
Integration and Differentiation 77ie Primitive 9 The Fundamental Theorem of Calculus
306
Mean Value Theorems of Integral Calculus
311
Integration by Parts
316
Change of Variable in an Integral
318
Second Mean Value Theorem
319
THE R1EMANNSTIELTJES INTEGRAL 1 Definitions and Existence of the Integral
330
A Condition of Integrability
333
Some Theorems
334
A Definition Integral as a limit of sum
338
Some Important Theorems
346
IMPROPER INTEGRALS 1 Introduction
351
Infinite Range of Integration
370
Integrand as a Product of Functions
389
UNIFORM CONVERGENCE 1 Pointwise Convergence
404
Uniform Convergence on an Interval
406
Tests for Uniform Convergence
412
Properties of Uniformly Convergent Sequences and Series
422
FUNCTIONS OF SEVERAL VARIABLES
492
Continuity
501
Partial Derivatives
505
Differentiability
509
Partial Derivatives of Higher Order
517
Differentials of Higher Order
524
Functions of Functions
526
Change of Variables
533
Taylors Theorem
544
Maxima and Minima
548
Functions of Several Variables
554
IMPLICIT FUNCTIONS
562
Jacobians
567
Stationary Values under Subsidiary Conditions
575
INTEGRATION ON R2
588
330
594
Double Integrals
596
Double Integrals Over a Region
618
Greens Theorem
629
Change of Variables
637
INTEGRATION ON R8
652
Line Integrals
657
Surfaces
662
Surface Integrals
670
Stokes Theorem First generalization of Greens Theorem
687
The Volume of a Cylindrical Solid by Double Integrals
692
Volume Integrals Triple Integrals
698
Gausss Theorem Divergence Theorem
708
METRIC SPACES
726
Open and Closed Sets
737
Convergence and Completeness
758
Continuity and Uniform Continuity
768
Compactness
781
Connectedness
800
THE LEBESGUE INTEGRAL
811
Sets of Measure Zero
820
Borel Sets
824
Measurable Functions
828
Measurability of the Sum Difference Product and Quotient Measurable Functions
831
Lebesgue Integral
836
Properties of Lebesgue Integral for Bounded Measurable Functions
839
Lebesgue Integral of a Bounded Function Over a Set of Finite Measure
845
Lebesgue Integral for Unbounded Functions
850
The General Integral
853
Lebesgue Theorem on Bounded Convergence
857
Integrability and Measurability
859
Lebesgue Integral on Unbounded Sets or Intervals
869
BETA AND GAMMA FUNCTIONS
872
Order in R
885
251
899
355
902
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Real Analysis
N. L. Carothers
Limited preview - 2000
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