Amazing and Aesthetic Aspects of Analysis PDF
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Paul Loya Amazing and Aesthetic Aspects of Analysis Amazing and Aesthetic Aspects of Analysis Paul Loya Amazing and Aesthetic Aspects of Analysis 123 PaulLoya Department ofMathematics Binghamton University Binghamton, NY USA ISBN978-1-4939-6793-3 ISBN978-1-4939-6795-7 (eBook) https://doi.org/10.1007/978-1-4939-6795-7 LibraryofCongressControlNumber:2016958470 ©PaulLoya2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerScience+BusinessMedia,LLC partofSpringerNature Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A. Preface I have truly enjoyed writing this book on some amazing and aesthetic aspects of analysis. Admittedly, some of the writing is too overdone (e.g., overuse of collo- quiallanguageandabundantalliterationattimes).ButwhatcanIsay?Iwashaving fun.ThesectionsofthebookwithanHaremeanttobeoptionalorjustforfunand don’t interfere with other sections, besides perhaps other starred sections. Most of the quotations that you’ll find in these pages are taken from the website http://www-gap.dcs.st-and.ac.uk/history/Quotations/. The contents of this book are based on lectures I have given to Binghamton Universitystudentstakingourfallsemesterundergraduaterealanalysiscoursefrom 2003 to about 2006. The audience consisted of math majors, including actuarial students, as well as students from the fields of chemistry, computer science, eco- nomics, and physics, among others. In order to interest such a diverse body of students, I wanted to write a book that not only teaches the fundamentals of analysis, but also shows its usefulness, beauty, and excitement. I also wanted a book that is personal, in which the students come with me on a journey through some amazing and aesthetic aspects of analysis. There are no derivatives or inte- grals in this book. This is on purpose, because I wanted to focus on the “ele- mentary” limiting processes only, those directly involving sequences and continuity, without the “higher” technology of calculus. The student completing Chapters 1 through 4 of this book will have mastered the fundamental arts of analysis and can move on to the “higher” arts, like the Lebesgue theory of inte- gration, which I usually teach in our spring semester real analysis course. Besides giving an appreciation of the amazing and aesthetic aspects of analysis (of course!), the overarching goals of this textbook are similar to those of any advanced math textbook, regardless of the subject: GOALS OF THIS TEXTBOOK. THE STUDENT WILL BE ABLE TO… (cid:129) Comprehend and write mathematical reasonings and proofs. (cid:129) Wield the language of mathematics in a precise and effective manner. (cid:129) State the fundamental ideas, axioms, definitions, and theorems on which real analysis is built and flourishes. v vi Preface (cid:129) Articulate the need for abstraction and the development of mathematical tools and techniques in a general setting. The objectives of this book make up the framework of how these goals will be accomplished, and more or less follow the chapter headings: OBJECTIVES OF THIS TEXTBOOK. THE STUDENT WILL BE ABLE TO… (cid:129) Identify the interconnections between set theory and mathematical statements and proofs (Chapter 1). (cid:129) State the fundamental axioms of the natural, integer, and real number systems and how the completeness axiom of the real number system distinguishes that system from the rational system in a powerful way (Chapter 2). (cid:129) Apply the rigorous e-N definition of convergence for sequences and series and recognize monotone and Cauchy sequences (Chapter 3). (cid:129) Applytherigorouse-–definitionoflimitsandcontinuityforfunctionsandapply the fundamental theorems of continuous functions (Chapter 4). (cid:129) Analyze the convergence properties of an infinite series, product, or continued fraction (mainly Chapters 5–8). (cid:129) Identify series, product, and continued fraction formulas for the various ele- mentary functions and constants (Throughout!). In one semester, I usually review parts of Chapters 1 and 2, then cover most of Chapters 3 and 4, and end with some applications from Chapters 5–8. Although not a history book (though I do give tiny history bites throughout the book) nor a “little” book like Herbert Westren Turnbull’s book The Great Mathematicians, in the words of Turnbull, I do hope … Ifthislittlebookperhapsmaybringtosome,whoseacquaintancewithmathematicsisfull oftoilanddrudgery,aknowledgeofthosegreatspiritswhohavefoundinitaninspiration anddelight,thestoryhasnotbeentoldinvain.Thereisalargenessaboutmathematicsthat transcendsraceandtime:mathematics mayhumblyhelp inthemarket-place, butitalso reachestothestars.Toone,mathematicsisagame(butwhatagame!)andtoanotheritis the handmaiden of theology. The greatest mathematics has thesimplicity and inevitable- nessofsupremepoetryandmusic,standingontheborderlandofallthatiswonderfulin Science,andallthatisbeautifulinArt.Mathematicstransfiguresthefortuitousconcourse ofatomsintothetraceryofthefingerofGod. HerbertWestrenTurnbull(1885–1961).Quotedfrom(243,p.141) I’dliketothankBrettBernsteinandYeLiforlookingoverthenotesandgiving manyvaluablesuggestions,withspecialthankstoYeLiforwritingupsolutionsto many problems (which will eventually be available as a student and instructor’s guide)andforpushingmetofinallygetthisbookintoprint.ThanksalsotoDikran KaragueuzianandDennisPixtonforusingthebookwhentheytaughtrealanalysis. The editors at Springer have been wonderful to work with; my thanks to them all. Also,thankstothemanypeoplethroughouttheworldwhohaveemailedmeabout the book with encouragement and comments, including Jeremiah Goertz, Scott Lindstrom,ZbigniewSzewczakandFabioRicci.TherearemanyotherstowhomI Preface vii owethanks,butduetosituationsbeyondmycontrol,I’veeitherlosttheiremailsor was not able to reply. Please accept my sincerest gratitude to you all. I thank my wife, Deborah, as well as my children, Melodie, Blaise, Theo, and Harmonie, for their continued support. Amid the difficulties, I thank and dedicate this book to Jesus, my Lord, Savior, and friend, for allowing me to complete this work. Soli Deo Gloria Binghamton, NY, USA Paul Loya A Word to the Student One can imagine mathematics as a movie with exciting scenes, action, plots, etc. Thereareacouplethingsyoucando.First,youcansimplysitbackandwatchthe movie playing out. Second, you can take an active role in shaping the movie. A mathematician does both at times, but is more an actor than an observer. I recommend that you be an actor in the great mathematics movie. To do so, I recommend that you read this book with a pencil and paper athand, writing down definitions,workingthroughexamples,fillinginanymissingdetails,andofcourse doingexercises(eventheonesthatarenotassigned).1Ofcourse,pleasefeelfreeto mark up the book as much as you wish with remarks and highlighting and even corrections if you find a typo or error. (Just let me know if you find one!) The sections with an H are optional and are meant to showcase some of the most breathtaking scenes in this Amazing and Aesthetic Aspects of Analysis. 1There are many footnotes in this book. Most are quotations from famous mathematicians and othersareremarksthatImightmaketoyouifIwerereadingthebookwithyou.Allfootnotesmay beignoredifyouwish! ix Some of the Most Beautiful Formulas in the World Inadditiontotheformulasinvolving…2=6onthefrontpageofthisbook,beloware moreofthemaincharacterswe’llmeetonourjourney,U(thegoldenratio),log2, pffiffiffi …,(cid:2)(theEuler–Mascheroniconstant),‡ðzÞ(thezetafunction), 2,ande.Indicated are a section and page number where we prove the formula, most of which are proved in different ways on other pages. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi uuu vuuffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 5 t t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U¼ ¼ 1þ 1þ 1þ 1þ 1þ 1þ (cid:2)(cid:2)(cid:2) (Section 3.3, p. 177) 2 1 U¼1þ (Section 3.4, p. 193) 1 1þ 1 1þ . . 1þ . pffiffiffi 1 2¼1þ (Section 3.4, p. 192) 1 2þ 1 2þ . . 2þ . 1 1 1 1 1 log2¼1(cid:3) þ (cid:3) þ (cid:3) þ (cid:2)(cid:2)(cid:2) (Section 4.7, p. 311) 2 3 4 5 6 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffi u 2 1 1 1 1 t1 1 1 1 1 ¼ (cid:2) þ (cid:2) þ þ (cid:2)(cid:2)(cid:2) (Section 5.1, p. 382) … 2 2 2 2 2 2 2 2 2 … 1 2 2 4 4 6 6 8 ¼ (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2)(cid:2) (Section 5.1, p. 389) 2 1 1 3 3 5 5 7 7 xi xii SomeoftheMostBeautifulFormulasintheWorld (cid:3) (cid:4)(cid:3) (cid:4)(cid:3) (cid:4)(cid:3) (cid:4) 1 1 1 1 1þ 1þ 1þ 1þ (cid:2)(cid:2)(cid:2) 1(cid:2)3 3(cid:2)5 5(cid:2)7 7(cid:2)9 …¼ (Section 5.1, p. 390) 1 1 1 1 þ þ þ þ (cid:2)(cid:2)(cid:2) 1(cid:2)3 3(cid:2)5 5(cid:2)7 7(cid:2)9 … 1 1 1 1 ¼ (cid:3) þ (cid:3) þ (cid:2)(cid:2)(cid:2) (Section 5.2, p. 400) 4 1 3 5 7 ‡ð2Þ ‡ð3Þ ‡ð4Þ ‡ð5Þ (cid:2) ¼ (cid:3) þ (cid:3) þ (cid:2)(cid:2)(cid:2) (Section 6.8, p. 517) 2 3 4 5 2 2 2 2 log2¼1þpffi2ffiffi(cid:2)1þpffipffiffiffiffi2ffiffiffiffiffi(cid:2)1þqffipffiffiffiffiffipffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffi(cid:2)1þrffiqffiffiffiffiffipffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffi(cid:2)(cid:2)(cid:2)(Section7.1,p.538) (cid:3) (cid:4)(cid:3) (cid:4)(cid:3) (cid:4)(cid:3) (cid:4)(cid:3) (cid:4) z2 z2 z2 z2 z2 sin…z¼…z 1(cid:3) 1(cid:3) 1(cid:3) 1(cid:3) 1(cid:3) (cid:2)(cid:2)(cid:2)(Section7.3,p.547) 12 22 32 42 52 pffiffiffi 2 2 6 6 10 10 2¼ (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2)(cid:2) (Section 7.3, p. 554) 1 3 5 7 9 11 … 1 2z 2z 2z 2z ¼ (cid:3) þ (cid:3) þ (cid:3)(cid:2)(cid:2)(cid:2) (Section 7.4, p. 560) sin…z z z2(cid:3)12 z2(cid:3)22 z2(cid:3)32 z2(cid:3)42 2z 3z 5z 7z 11z 13z 17z ‡ðzÞ¼ (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2)(cid:2)(Section7.6,p.566) 2z(cid:3)1 3z(cid:3)1 5z(cid:3)1 7z(cid:3)1 11z(cid:3)1 13z(cid:3)1 17z(cid:3)1 1 log2¼ (Section 8.2, p. 600) 12 1þ 22 1þ 32 1þ 42 1þ . . 1þ . 4 12 ¼1þ (Section 8.2, p. 602) … 32 2þ 52 2þ 72 2þ . . 2þ . 2 e¼1þ1þ (Section 8.2, p. 607) 3 2þ 4 3þ 5 4þ . . 5þ .
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