Introduction to Calculus and Classical Analysis

434 Pages·2016·2.567 MB·English

by  Omar Hijab

Undergraduate Texts in Mathematics
Omar Hijab
Introduction
to Calculus
and Classical
Analysis
Fourth Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
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SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
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RogerE.Howe,YaleUniversity
MichaelOrrison,HarveyMuddCollege
LisetteG.dePillis,HarveyMuddCollege
JillPipher,BrownUniversity
FadilSantosa,UniversityofMinnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
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illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Moreinformationaboutthisseriesathttp://www.springer.com/series/666
Omar Hijab
Introduction to Calculus
and Classical Analysis
Fourth Edition
123
OmarHijab
CollegeofScienceandTechnology
TempleUniversity
Philadelphia,PA,USA
ISSN0172-6056 ISSN2197-5604 (electronic)
UndergraduateTextsinMathematics
ISBN978-3-319-28399-9 ISBN978-3-319-28400-2 (eBook)
DOI10.1007/978-3-319-28400-2
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To M.A.W.

Preface
For undergraduate students, the transition from calculus to analysis is often
disorienting and mysterious. What happened to the beautiful calculus for-
mulas?Where did (cid:2)-δ andopensets come from?Itis notuntil later thatone
integratesthese seemingly distinct points ofview. When teaching “advanced
calculus,” I always had a difficult time answering these questions.
Now,everymathematicianknowsthatanalysisarosenaturallyinthenine-
teenthcenturyoutofthecalculusoftheprevioustwocenturies.Believingthat
itwaspossibletowriteabookreflecting,explicitly,thisorganicgrowth,Iset
out to do so.
Ichoseseveralofthejewelsofclassicaleighteenth-andnineteenth-century
analysisand inserted them near the end of the book, insertedthe axioms for
reals at the beginning, and filled in the middle with (and only with) the
material necessaryfor clarity and logical completeness. In the process, every
little piece of one-variablecalculus assumedits proper place, and theory and
application were interwoven throughout.
Letmedescribesomeoftheunusualfeaturesinthistext,asthereareother
booksthatadopttheabovepointofview.Firstisthesystematicavoidanceof
(cid:2)-δ arguments. Continuous limits are defined in terms of limits of sequences,
limits ofsequences aredefined in terms ofupper andlowerlimits, and upper
andlowerlimitsaredefinedintermsofsupandinf.Everybodythinksinterms
ofsequences,sowhydoweteachourundergraduates(cid:2)-δ’s?(Incalculustexts,
especially, doing this is unconscionable.)
Thesecondfeatureisthetreatmentofintegration.Wefollowthestandard
treatment motivated by geometric measure theory, with a few twists thrown
in: The area is two-dimensional Lebesgue measure, defined on all subsets of
R2, the integral of an arbitrary1 nonnegative function is the area under its
graph, and the integral of an arbitrary integrable function is the difference
of the integrals of its positive and negative parts.
1 Notnecessarily measurable.
vii
viii Preface
IndealingwitharbitrarysubsetsofR2 andarbitraryfunctions,onlyafew
basicpropertiescanbe derived;nevertheless,surprisingresults areavailable;
for example, the integral of an arbitrary integrable function over an interval
is a continuous function of the endpoints.
Arbitrary functions are considered to the extent possible not because of
generalityforgenerality’ssake,butbecausetheyfitnaturallywithinthecon-
textlaidouthere.For example,the density theoremandmaximalinequality
are valid for arbitrarysets and functions, and the first fundamental theorem
is valid for an integrable f if and only if f is measurable.
In Chapter 4 we restrict attention to the class of continuous functions,
which is broad enough to handle the applications in Chapter 5, and derive
the second fundamental theorem in the form
(cid:2)
b
f(x)dx=F(b−)−F(a+).
a
Herea,b,F(a+)orF(b−)maybeinfinite,extendingtheimmediateapplica-
bility, andthe continuousfunction f need onlybe nonnegativeor integrable.
Thethirdfeatureisthetreatmentofthetheoremsinvolvinginterchangeof
limits and integrals. Ultimately, all these theorems depend on the monotone
convergence theorem which, from our point of 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, differentiation under the integral sign is used to compute the Gaussian
integral.
Thetreatmentofintegrationpresentedhereemphasizesgeometricaspects
rather than technicality. The most technical aspects, the derivation of the
MethodofExhaustionin§4.5,maybe skippeduponfirstreading,orskipped
altogether, without affecting the flow.
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
presentedinacomplexsettingusingtranscendentaltechniques.Forexample,
included is:
(cid:129) A real-variable derivation of Gauss’ AGM formula motivated by the unit
circle map √ √
ax+i by
x(cid:2)+iy(cid:2) = √ √ .
ax−i by
(cid:129) A real-variable computation of the radius of convergence of the Bernoulli
series, derived via the infinite product expansion of sinhx/x, which is in
turn derived by combinatorial real-variable methods.
Preface ix
(cid:129) The zeta functional equation is derived via the theta functional equation,
which is in turn derived via the connection to the parametrization of the
AGM curve.
The fifth 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 in §1.4 to exp(−ζ(cid:2)(0))= 2π
in §5.8.
Because we take the real numbers as our 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, while the
trigonometricfunctions are initially defined via their Taylor series,later it is
shown how they may be defined via the unit circle.
Although it is helpful for the readerto haveseencalculus prior to reading
this text, the developmentdoes not presume this. We feel it is important for
undergraduatestosee,atleastonceintheirfouryears,anonpedantic,purely
logicaldevelopmentthatreallydoesstartfromscratch(ratherthanpretends
to), is self-contained, and leads to nontrivial and striking results.
Applications include a specific transcendental number; convexity,elemen-
tary symmetric polynomial inequalities, subdifferentials, and the Legendre
transform;Machin’sformula;the Cantorset;the Bailey–Borwein–Plouffese-
ries (cid:4) (cid:5)
(cid:3)∞
1 4 2 1 1
π = − − − ;
16n 8n+1 8n+4 8n+5 8n+6
n=0
continuedfractions;LaplaceandFouriertransforms;Besselfunctions;Euler’s
constant;theAGMiteration;thegammaandbetafunctions;Stirlingidentity
(cid:4)(cid:2) (cid:5)
s+1 √
exp logΓ(x)dx =sse−s 2π, s>0;
s
theentropyofthebinomialcoefficients;infinite productsandBernoullinum-
bers; theta functions and the AGM curve; the zeta function; the zeta series
x2 x3 x4
log(x!)=−γx+ζ(2) −ζ(3) +ζ(4) −..., 1≥x>−1;
2 3 4
primesinarithmeticprogressions;theEuler–Maclaurinformula;andtheStir-
ling series.
After the applications, Chapter 6 develops the fundamental theorems in
their general setting, basedon the sunriselemma. This materialis ata more
advanced level, but is included to point the reader towardtwentieth-century
developments.