Integral: An Easy Approach after Kurzweil and Henstock,9780521779685

Integral: An Easy Approach after Kurzweil and Henstock

by
ISBN13: 9780521779685
ISBN10: 0521779685
Format: Paperback
Pub. Date: 2000-04-28
Publisher(s): Cambridge University Press
  • Free Shipping Icon

    This Item Qualifies for Free Shipping!*

    *Excludes marketplace orders.

List Price: $60.89

Buy New

Arriving Soon. Will ship when available.
$57.99
Add to Cart

Rent Textbook

Select for Price
Add to Cart
There was a problem. Please try again later.

Used Textbook

We're Sorry
Sold Out

eTextbook

We're Sorry
Not Available

Buy from our Marketplace starting at $94.02

How Marketplace Works:

  • This item is offered by an independent seller and not shipped from our warehouse
  • Item details like edition and cover design may differ from our description; see seller's comments before ordering.
  • Sellers much confirm and ship within two business days; otherwise, the order will be cancelled and refunded.
  • Marketplace purchases cannot be returned to eCampus.com. Contact the seller directly for inquiries; if no response within two days, contact customer service.
  • Additional shipping costs apply to Marketplace purchases. Review shipping costs at checkout.

Summary

Integration has a long history: its roots can be traced as far back as the ancient Greeks. The first genuinely rigorous definition of an integral was that given by Riemann, and further (more general, and so more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory, and it presents a unified yet elementary approach that is suitable for beginning graduate and final year undergraduate students.

Table of Contents

Preface viii
List of Symbols
xi
Introduction
1(21)
Historical remarks
1(2)
Notation and the Riemann definition
3(3)
Basic theorems, upper and lower integrals
6(4)
Differentiability, continuity and integrability
10(6)
Limit and R-integration
16(2)
Exercises
18(4)
Basic Theory
22(54)
Introduction
22(1)
Motivation
22(3)
Cousin's lemma
25(4)
Applications of Cousin's lemma
26(3)
The definition
29(3)
Basic theorems
32(14)
The Fundamental Theorem of calculus
46(4)
Consequences of the Fundamental Theorem
50(5)
Improper integrals
55(4)
Integrals over unbounded intervals
59(5)
Alternative approach to integration over unbounded intervals
64(2)
Negligible sets
66(3)
Complex valued function
69(2)
Exercises
71(5)
Development of the Theory
76(75)
Equivalent forms of the definition
76(5)
Henstock's lemma
81(2)
Functions of bounded variation
83(3)
Absolute integrability
86(2)
Limit and KH-integration
88(12)
Absolute continuity
100(4)
Equiintegrability
104(6)
The second mean value theorem
108(2)
Differentiation of integrals
110(2)
Characterization of the KH-integral
112(3)
Lebesgue points, approximation by step functions
115(2)
Measurable functions and sets
117(10)
A non-measurable set
126(1)
The McShane integral
127(8)
A short proof
135(1)
The Lebesgue integral
135(5)
F. Riesz' definition
139(1)
Quick proofs
140(1)
Differentiation almost everywhere
140(5)
Exercises
145(6)
The SL-integral
151(24)
The strong Luzin condition
151(4)
SL-integration
155(7)
Limit and SL-integration
162(7)
Equivalence with the KH-integral
169(2)
Exercises
171(4)
Generalized AC Functions
175(27)
Prologue
175(2)
Uniformly AC functions
177(2)
AC* and VB* on a set
179(8)
ACG* functions
187(3)
Controlled convergence
190(7)
Exercise
197(5)
Integration in Several Dimensions
202(50)
Introduction
202(1)
Sets in Rn
203(1)
Divisions, partitions
204(6)
The definition
210(5)
Basic theorems
215(5)
Prelude to Fubini's theorem
217(3)
Other theorems in Rn
220(9)
Negligible sets
220(1)
Henstock's lemma
220(1)
Absolute integrability
221(2)
Convergence, measurability, AC
223(6)
The Fubini theorem
229(7)
Change of variables
236(12)
Introductory examples
236(4)
Notation, lemmas
240(1)
The theorem
241(7)
Exercises
248(4)
Some Applications
252(47)
Introduction
252(1)
A line integral
253(26)
Green's theorem
267(7)
The Cauchy theorem
274(5)
Differentiation of series
279(3)
Dirichlet's problem and the Poisson integral
282(4)
Summability of Fourier series
286(3)
Fourier series and the space L2
289(6)
Exercises
295(4)
Appendix 1 Supplements 299(6)
Bibliography 305(3)
Index 308

An electronic version of this book is available through VitalSource.

This book is viewable on PC, Mac, iPhone, iPad, iPod Touch, and most smartphones.

By purchasing, you will be able to view this book online, as well as download it, for the chosen number of days.

Digital License

You are licensing a digital product for a set duration. Durations are set forth in the product description, with "Lifetime" typically meaning five (5) years of online access and permanent download to a supported device. All licenses are non-transferable.

More details can be found here.

A downloadable version of this book is available through the eCampus Reader or compatible Adobe readers.

Applications are available on iOS, Android, PC, Mac, and Windows Mobile platforms.

Please view the compatibility matrix prior to purchase.