1、Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet For other titles Published in this series, go to Omar Hijab and Classical Analysis Introduction to Calculus ThirdEditionAll rights reserved. 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis
2、. Use in connection or dissimilar methodology now known or hereafter developed is forbidden. to proprietary rights. Printed on acid-free paper not identied as such, is not to be taken as an expression of opinion as to whether or not they are subject with any form of information storage and retrieval
3、, electronic adaptation, computer software, or by similarSpringer New York Dordrecht Heidelberg London Department of Mathematics USA Editorial Board K.A. Ribet Mathematics Department San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 U
4、SA USA axlersfsu.edu ribetmath.berkeley.edu This work may not be translated or copied in whole or in part without the written ISSN 0172-6056 S. Axler OmarHijab Temple University Mathematics Department Philadelphia, PA 19122 hijabtemple.edu ISBN 978-1-4419-9487-5 e-ISBN 978-1-4419-9488-2 DOI 10.1007/
5、978-1-4419-9488-2 The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are SpringerScience+BusinessMedia,LLC2011 permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY Mathematics Subject Classification (20
6、10): 41XX,40XX,33XX,05XX Library of Congress Control Number: 2011923653 SpringerispartofSpringerScience+BusinessMedia()To M.A.W.Preface For undergraduate students, the transition from calculus to analysis is often disorientingandmysterious.Whathappenedtothe beautiful calculusformu- las? From where d
7、id and open sets come? It is not until later that one integrates these seemingly distinct points of view. When teaching advanced calculus“ I always had a dicult time answering these questions. Now,everymathematicianknowsthatanalysisarosenaturallyinthenine- teenthcenturyoutofthecalculusoftheprevioust
8、wocenturies.Believingthat it waspossible to write a book re ecting,explicitly,this organicgrowth,I set out to do so. I chose several of the jewels of classical eighteenth and nineteenth century analysis and inserted them at the end of the book, inserted the axioms for reals at the beginning, and lle
9、d in the middle with (and only with) the material necessary for clarity and logical completeness. In the process, every little piece of one-variable calculus assumed its proper place, and theory and application were interwoven throughout. Letmedescribesomeoftheunusualfeaturesinthistext,asthereareoth
10、er booksthatadopttheabovepointofview.Firstisthesystematicavoidanceof arguments. Continuous limits are de ned in terms of limits of sequences, limits of sequences are de ned in terms of upper and lower limits, and upper andlowerlimitsarede nedintermsofsupandinf.Everybodythinksinterms ofsequences,sowh
11、ydoweteachourundergraduates s?(Incalculustexts, especially, doing this is unconscionable.) The secondfeature is the treatmentofintegration.We followthe standard treatment motivated by geometric measure theory, with a few twists thrown in: the area is two-dimensional Lebesgue measure, de ned on all s
12、ubsets of R 2 , the integral of an arbitrary nonnegative function is the area under its graph, and the integral of an arbitrary integrable function is the di erence of the integrals of its positive and negative parts. Indealingwitharbitrarysubsets ofR 2 andarbitraryfunctions, onlya few basic propert
13、ies can be derived; nevertheless, surprising results are available, for example, the integral of an arbitrary integrable function over an interval viiviii Preface is a continuous function of the endpoints. Arbitrary functions are considered to the extent possible not because of generality for genera
14、litys sake, but because they t naturally within the context laid out here. After this, we restrictattentionto the class ofcontinuousfunctions, which is broad enough to handle the applications in Chapter 5, and derive the fundamental theorem in the formb a f(x)dx =F(b)F(a+); herea,b,F(a+),orF(b)maybe
15、in nite,broadeningtheimmediateapplica- bility, and the continuous functionf need only be nonnegative or integrable. Thethirdfeatureisthetreatmentofthetheoremsinvolvinginterchangeof limits and integrals. Ultimately, all these theorems depend on the monotone convergence theorem which, from our point o
16、f view, follows from the Greek mathematicians method of exhaustion. Moreover, these limit theorems are stated only after a clear and nontrivial need has been elaborated. For exam- ple, di erentiation under the integral sign is used to compute the Gaussian integral. As a consequence of our treatment
17、of integration, we can dispense with uniform convergence and uniform continuity. (If the reader has any doubts about this, a glance at the range of applications in Chapter 5 will help.) Nevertheless, we give a careful treatment of uniform continuity, and use it in the exercises to discuss an alterna
18、te de nition of the integral that was important in the nineteenth century (the Riemann integral). The treatment of integration also emphasizes geometric aspects rather than technicality; the most technical aspect, the derivation 1 in 4.5 of the method of exhaustion, may be skipped upon rst reading,
19、or skipped alto- gether, without a ecting the ow. The fourth feature is the use of real-variable techniques in Chapter 5. We do this to bring out the elementary nature of that material, which is usually presented in a complex setting using transcendental techniques. For example,included is a real-va
20、riablecomputation ofthe radius ofconvergence of the Bernoulliseries,derivedvia the in nite product expansionofsinhx/x, whichis inturnderivedbycombinatorialreal-variablemethods,andthe zeta functional equation is derived via the theta functional equation, which is in turn derived via the connection to
21、 the parametrization of the AGM curve. The fth feature is our emphasis on computational problems. Computa- tion, here, is often at a deeper level than expected in calculus courses and varies from the high school quadratic formula in1.4 to (0) =log(2)/2 in5.8. Because we take the real numbers as our
22、starting point, basic facts about the natural numbers, trigonometry, or integration are rederived in this con- text, either in the body of the text or as exercises.For example,although the 1 Measurable subsets of R 2 make a brief appearance in 4.5.Preface ix trigonometric functions are initially de
23、ned via their Taylor series, later it is shown how they may be de ned via the unit circle. Although it is helpful for the reader to have seen calculus prior to reading this text, the development does not presume this. We feel it is important for undergraduatestosee,atleastonceintheirfouryears,anonpe
24、dantic,purely logicaldevelopmentthat really does start fromscratch(rather than pretends to), is self-contained, and leads to nontrivial and striking results. We have attempted to present applications from many parts of analysis, many of which do not usually make their way into advanced calculus book
25、s. For example, we discuss a speci c transcendental number; convexity, subd- i erentials, and the Legendre transform; Machins formula; the Cantor set; the BaileyBorweinPlou e series; continued fractions; Laplace and Fourier transforms;Besselfunctions;Eulersconstant;theAGMiteration;thegamma and beta
26、functions; the entropy of the binomial coecients; in nite products and Bernoulli numbers; theta functions and the AGM curve; the zeta func- tion; primes in arithmetic progressions; the EulerMaclaurin formula; and the Stirling series. As an aid to self-study and assimilation, there are 385 problems w
27、ith all solutions at the back of the book. Every exercise can be solved using only previous material from this book. If some of the more technical parts (such as 4.5) are skipped, this book is suitable for a one-semester course (the extent to which this is possible depends on the students calculus a
28、bilities). Alternatively, thoroughly covering the entire text lls up a yearcourse, as I have done at Temple teaching our advanced calculus sequence. Philadelphia, Fall 2010 Omar HijabContents Preface . vii 1 The Set of Real Numbers 1 1.1 Sets and Mappings 1 1.2 The Set R. 4 1.3 The Subset N and the
29、Principle of Induction 9 1.4 The Completeness Property . 16 1.5 Sequences and Limits 20 1.6 Nonnegative Series and Decimal Expansions . 31 1.7 Signed Series and Cauchy Sequences . 37 2 Continuity 47 2.1 Compactness. 47 2.2 Continuous Limits 48 2.3 Continuous Functions . 52 3 Di erentiation . 69 3.1
30、Derivatives 69 3.2 Mapping Properties. 77 3.3 Graphing Techniques . 83 3.4 Power Series . 94 3.5 Trigonometry . 107 3.6 Primitives . 114 4 Integration . 123 4.1 The Cantor Set 123 4.2 Area 127 4.3 The Integral 143 4.4 The Fundamental Theorem of Calculus . 160 4.5 The Method of Exhaustion 173 xixii C
31、ontents 5 Applications 185 5.1 Eulers Gamma Function 185 5.2 The Number . 191 5.3 Gauss ArithmeticGeometric Mean (AGM) . 206 5.4 The Gaussian Integral. 214 5.5 Stirlings Approximation ofn! 224 5.6 In nite Products . 231 5.7 Jacobis Theta Functions 241 5.8 Riemanns Zeta Function . 248 5.9 The EulerMa
32、claurin Formula . 257 A Solutions . 267 A.1 Solutions to Chapter 1 267 A.2 Solutions to Chapter 2 283 A.3 Solutions to Chapter 3 290 A.4 Solutions to Chapter 4 310 A.5 Solutions to Chapter 5 330 References 357 Index. 359Chapter 1 The Set of Real Numbers A Note to the Reader This text consists of man
33、y assertions, some big, some small, some almost insigni cant. These assertions are obtained from the properties of the real numbers by logical reasoning. Assertions that are especially important are called theorems. An assertions importance is gauged by many factors, in- cluding its depth, how many
34、other assertions it depends on, its breadth, how many other assertions are explained by it, and its level of symmetry. The later portions of the text depend on every single assertion, no matter how small, made in Chapter 1. The text is self-contained, and the exercises are arranged in order: Every e
35、xercise can be done using only previous material from this text. No outside material is necessary. Doing the exercises is essential for understanding the material in the text. Sections are numbered sequentially within each chapter; for example, 4.3 means the third section in Chapter 4. Equation numb
36、ers are written within parenthesesandexercisenumbersinbold.Theorems,equations,andexercises are numbered sequentially within each section; for example, Theorem 4.3.2 denotes the second theorem in4.3, (4.3.1) denotes the rst numbered equa- tion in4.3, and 4.3.3 denotes the third exercise at the end of
37、4.3. Throughout, we use the abbreviation i to mean if and only if and to signal the end of a derivation. 1.1 Sets and Mappings Weassumethereaderisfamiliarwiththeusualnotionsofsetsandmappings, but we review them to x the notation. DOI 10.1007/978-1-4419-9488-2_1, Springer Science+Business Media, LLC 2011 1 O. Hijab, Introduction to Calculus and Classical Analysis, Undergraduate Texts in Mathematics,
