Amazing and Aesthetic Aspects of Analysis

730 Pages·2018·10.383 MB·English

by  Paul Loya

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
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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 .