INTRODUCTIONTOCALCULUS ANDANALYSISVolumeOneOther Books bytheSameAuthorsRichard CourantDifferential and Integral Calculus, Interscience Publishers,Vol.I, second edition, 1937;Vol.II, first edition, 1936Dirichlet's Principles, Conformal Mappingand MinimalSurfaces,Interscience Publishers, 1950Methods of Mathematical Physics (and D. Hilbert), IntersciencePublishers, Vol.1,1953;Vol. 11,1962;Vol.III, in press.Supersonic Flow and Shock Waves (andK.O. Friedrichs),Interscience Publishers, 1948FritzJohnPartial Differential Equations (andL. Bers andM.Schechter),Interscience Publishers, 1964Plane Waves and Spherical MeansApplied to Partial DifferentialEquations, Interscience Publishers, 1955IntroductiontoCALCULUSANDANALYSISVolumeOneRichard Courant and FritzJohnCourantInstitute of Mathematical SciencesNew YarkUniversityInterscience PublishersADivision of JohnWileyand Sons, Inc.New York London SydneyCopyright 1965 byRichard CourantAll Rights Reserved. This bookoranypart thereof must notbereproducedinanyformwithout the written permissionofthepublisher.Library of CongressCatalogCard Number:65-16403Printed in theUnited States of AmericaPrefaceDuringthe latter part ofthe seventeenthcenturythe newmathe-matical analysis emerged as the dominating force in mathematics.It ischaracterizedbytheamazinglysuccessful operationwithinfiniteprocesses or limits. Twoofthese processes, differentiationandinte-gration, became the core ofthe systematic Differential and IntegralCalculus, often simplycalled"Calculus,"basic forall of analysis.The importance of the new discoveries and methods was immediatelyfelt andcausedprofoundintellectualexcitement. Yet, togainmasteryof thepowerful art appearedat first aformidabletask, for theavail-able publications were scanty, unsystematic, and often lacking inclarity. Thus, it was fortunate indeed for mathematics and sciencein general that leaders in the newmovement soon recognized thevital need for writing textbooks aimed at making the subject ac-cessible to apublic much larger than the very small intellectual elite oftheearlydays. Oneof thegreatest mathematiciansof moderntimes,LeonardEuler, established inintroductorybooksafirmtraditionandthese books of the eighteenth century have remained sources of inspira-tion until today, even thoughmuchprogress has been made in theclarification and simplification of thematerial.AfterEuler, one author after theother adheredtotheseparation ofdifferential calculus fromintegral calculus, thereby obscuring a keypoint, thereciprocity between differentiationandintegration. Only in1927whenthefirst editionof R. Courant'sGermanVorlesungentiberDifferential und Integralrechnung, appeared in the Springer-Verlagwasthis separationeliminatedandthecalculuspresentedas aunifiedsubject.Fromthat German book and its subsequent editions the presentwork originated. With the cooperation of James and Virginia McShauea greatly expanded and modified English edition of the "Calculus" w ~ sprepared and published by Blackie and Sons in Glasgow since 1934, andvvi Prefacedistributed in the United States in numerous reprintings by Inter-science-Wiley.During the years it became apparent that the need of college and uni-versityinstructionintheUnitedStatesmadearewriting of thisworkdesirable. Yet, itseemedunwisetotamperwiththeoriginal versionswhich haveremained andstill areviable.Instead of trying to remodel the existing work it seemed preferable tosupplement it byanessentiallynewbookinmanywaysrelatedtotheEuropeanoriginalsbut morespecificallydirectedat theneedsof thepresent andfuturestudents intheUnitedStates. Such aplanbecamefeasible when Fritz John, who had already greatly helped in the prepara-tion of the firstEnglish edition, agreedtowritethe new book togetherwithR. Courant.While it differs markedly informand content fromthe original,it isanimatedbythesameintention: Toleadthestudent directlytotheheart ofthesubject andtopreparehimfor activeapplicationof hisknowledge. It avoids the dogmatic style which conceals the motivationand the roots of the calculus in intuitive reality. To exhibit the interac-tion betweenmathematical analysis and its various applications andtoemphasizetheroleof intuitionremainsanimportantaimof thisnewbook. Somewhat strengthenedprecisiondoesnot, aswehope, inter-ferewith this aim.Mathematics presented as a closed, linearly ordered, system of truthswithout referencetooriginandpurposehasitscharmandsatisfiesaphilosophical need. But the attitude of introverted science is unsuitablefor students who seek intellectual independence rather than indoctrina-tion; disregardfor applications andintuitionleads toisolation andatrophyof mathematics. It seemsextremelyimportant that studentsand instructorsshouldbeprotected fromsmug purism.Thebookis addressedtostudents onvariouslevels, tomathema-ticians, scientists, engineers. Itdoesnot pretendtomakethesubjecteasy byglossingover difficulties, butrathertriestohelpthegenuinelyinterested reader by throwing light on the interconnections and purposesof thewhole.Insteadof obstructingtheaccess tothewealthof facts bylengthydiscussions of a fundamental nature we have sometimes postponed suchdiscussionsto appendices in thevarious chapters.Numerous examplesandproblemsaregivenat theendof variouschapters. Somearechallenging, someareevendifficult; mostof themsupplement thematerial inthetext. Inanadditional pamphlet morePreface viiproblems andexercises ofa routine character will be collected, andmoreover, answers or hints forthe solutions will be given.Manycolleagues and friends have been helpful. Albert A. Blanknotonlygreatly contributedincisiveandconstructive criticism, buthealsoplayeda major roleinordering, augmenting, andsiftingof theproblemsandexercises, andmoreoverheassumedthemainresponsi-bility for the pamphlet. Alan Solomon helped most unselfishlyandeffectively in all phases of thepreparationof thebook. Thanksis alsoduetoCharlotteJohn, Anneli Lax, R. Richtmyer, andotherfriends,including James and Virginia McShane.Thefirst volumeisconcernedprimarilywithfunctions of asinglevariable, whereas the second volume will discuss the more ramifiedtheories of calculus forfunctions of several variables.Afinal remarkshouldbeaddressedtothestudent reader. Itmightprovefrustratingtoattempt mastery of thesubject by studying such abook page by page following an even path. Only by selecting shortcutsfirst and returning time and againto the same questions and difficultiescanonegraduallyattainabetterunderstandingfromamoreelevatedpoint.An attempt was made to assist users of the book by marking with anasterisksomepassageswhichmight impedethereaderat hisfirst at-tempt. Alsosomeofthe moredifficult problems are markedbyanasterisk.Wehopethat thework inthepresent new formwill be useful to theyounggenerationof scientists. Weareawareof manyimperfectionsand we sincerely invite critical comment which might be helpful for laterimprovenlents.Richard CourantFritzJohnJune1965ContentsChapter 1Introduction 11.1 The Continuum of Numbers 1a. The System of NaturalNumbers andItsExtension. Counting andMeasuring, 1b. Real Numbers andNestedIntervals, 7c. Decimal Fractions. Bases Other ThanTen, 9 d. Definition of Neighborhood, 12e.Inequalities, 121.2 The Concept of Function 17a. Mapping-Graph, 18 b. Definition of theConcept of Functions of a ContinuousVariable. Domain andRange of aFunction, 21c. GraphicalRepresentation. MonotonicFunctions, 24 d. Continuity,31 e. TheIntermediate Value Theorem. InverseFunctions, 441.3 The Elementary Functions 47a. RationalFunctions, 47 b. AlgebraicFunctions, 49 c. Trigonometric Functions, 49d. The ExponentialFunction andtheLogarithm, 51 e. CompoundFunctions,Symbolic Products, Inverse Functions, 521.4 Sequences 551.5 Mathematical Inductionix57x Contents1.6 The Limit of a Sequence 601 1 18. an=~ , 6 1 b. a2m =;; a2m-l =2m,62c. an = n~ 1,63 d. a" = vp, 64e. an = a!', 65f. GeometricalIllustration of theLimits ofa!' and vp, 6S g.The Geometric Series, 67h. an = yr;;: 69 i. an = ~ - v;;: 691.7 Further Discussion of the Concept of Limit 70a ~ Definition of Convergence and Divergence, 70b. Rational Operations withLimits,71c. Intrinsic Convergence Tests. MonotoneSequences,73 d. Infinite Series andtheSummation Symbol, 7S e. The Number e, 77f.The Number 7r as aLimit, 801.8 The Concept of Limit for Functions of a Con-tinuous Variable 82a. SomeRemarks about the ElementaryFunctions, 86Supplements 87S.l Limits and the Number Concept 89a. The RationalNumbers, 89 b. RealNumbers Determined byNested Sequences ofRational Intervals, 90 c.Order,Limits, andArithmetic Operations forReal Numbers, 92d. Completeness of the Number Continuum.Compactness of Closed Intervals. ConvergenceCriteria,94 e. Least Upper Bound andGreatest Lower Bound, 97 f. Denumerabilityof theRationalNumbers, 988.2 Theorems on Continuous Functions8.3 Polar Coordinates8.4 Remarks on Complex NumbersPROBLEMS99101103106Contents xiChapter 2 The Fundamental Ideas of the Integraland Differential Calculus 1192.1 The Integral 120a.Introduction, 120 b. The IntegralasanArea, 121 c. AnalyticDefinition of theIntegral. Notations, 1222.2 Elementary Examples of Integration 128a. Integration of Linear Function,128b. Integration of x2, 130 c. Integration ofxQforIntegersa ~ -1,131 d. Integration ofxQforRational aOther Than-1,134e. Integration of sin xandcos x, 1352.3 Fundamental Rules of Integration 136a. Additivity, 136 b. Integralof aSum of aProduct with aConstant, 137 c. EstimatingIntegrals, 138, d. The MeanValue TheoremforIntegrals, 1392.4 The Integral as a Function of the Upper Limit(Indefinite Integral) 1432.5 Logarithm Definedby an Integral 145a. Definition of the Logarithm Function, 145b.TheAddition Theorem forLogarithms, 1472.6 Exponential Function and Powers 149a. The Logarithm of the Number e, 149b.The Inverse Function of the Logarithm.The Exponential Function,150c.The Exponential Function asLimit ofPowers, 152 d. Definition of ArbitraryPowers of Positive Numbers, 152e. Logarithmsto Any Base, 1532.7 The Integral of an Arbitrary Power of x 1542.8 The Derivative 155a. The Derivative andthe Tangent, 156b. The Derivative asaVelocity, 162xii Contentsc. Examples of Differentiation, 163 d. SomeFundamental Rules for Differentiation, 165e. Differentiability and Continuity of Functions,166 f. Higher Derivatives and TheirSignificance,169 g.Derivative and DifferenceQuotient. Leibnitz's Notation, 171 h. TheMean Value Theorem of Differential Calculus, 173i. Proof of the Theorem, 175 j. TheApproximation of Functions by LinearFunctions. Definition of Differentials, 179k. Remarks on Applications to the NaturalSciences, 1832.9 The Integral, the Primitive Function, and theFundamental Theorems of the Calculus 184a. The Derivative of the Integral, 184 b. ThePrimitive Function and Its Relation to theIntegral, 186 c. The Use of the PrimitiveFunction for Evaluation of Definite Integrals, 189d. Examples, 191Supplement The Existence of the Definite Integralof a Continuous Function 192PROBLEMS 196Chapter 3 TheTechniques of Calculus 201Part A Differentiation and Integration of theElementary Functions 2013.1 The Simplest Rules for Differentiation andThekAppllcations 201a. Rules for Differentiation, 201b. Differentiation of the Rational Functions, 204c. Differentiation of the TrigonometricFunctions, 20S3.2 The Derivative of the Inverse Function 2068. General Formula, 206 b. The Inverse ofthe nth Power; Lie nth Root, 210 c. TheInverse Trigonometric Functions-Contents xiiiMultivaluedness,210 d. The CorrespondingIntegral Formulas, 215 e. Derivative andIntegral of the Exponential Function, 2163.3 Differentiation of Composite Functions 217a.Definitions, 217 b. The ChainRule, 218c. The GeneralizedMean Value Theorem of theDifferential Calculus, 2223.4 SomeApplicationsof theExponentialFunction 223a. Definition of the Exponential Function byMeans of a Differential Equation, 223b. Interest Compounded Continuously.Radioactive Disintegration, 224 c.Coolingor Heating of a Body by a SurroundingMedium, 225 d. Variation of theAtmospheric Pressure with the Height abovethe Surface of the Earth, 226 e. Progress of aChemicalReaction, 227 f. Switching anElectric Circuit on or off, 2283.5 The Hyperbolic Functions 228a.Analytical Definition, 228 b. AdditionTheorems and Formulas for Differentiation 231c. The Inverse Hyperbolic Functions, 232d. Further Analogies, 2343.6 Maxima and Minima 2368. Convexity and Concavity of Curves, 236b. Maxima andMinima-Relative Extrema.Stationary Points, 2383.7 The Order of Magnitude of Functions 2488. The Concept of Order of Magnitude. TheSimplest Cases, 248 b. The Order ofMagnitude of the Exponential Function and ofthe Logarithm, 249 c. General Remarks, 251d. The Order of Magnitude of a Function in theNeighborhood of an Arbitrary Point, 252e. The Order of Magnitude (or Smallness) of aFunction Tending to Zero, 252 f. The HO"and "0"Notation for Orders of Magnitude, 253xiv ContentsAPPENDIX255A.l Some Special Functions 2558. The Function y = e1/x2, 255 b. TheFunction y = etlx,2S6 c. The Functiony=tanh1/x, 257 d. The Functiony = xtanhl/x, 258 e. The Functiony=x sinl/x, yeO) = 0, 259A.2 Remarks on the Differentiability of Functions 259Part B Techniques of Integration 2613.8 Table of Elementary Integrals 2633.9 The Method of Substitution 2638. The Substitution Formula. Integral of aComposite Function, 263 b. A SecondDerivation of the Substitution Formula, 268c. Examples. Integration Formulas, 2703.10Further Examples of the Substitution Method2713.11Integration by Parts 274a. General Formula, 274 b. Further Examplesof Integration by Parts, 276 c. IntegralFormula for (b) +I(a), 278 d. RecursiveFormulas, 278 lite.Wallis's Infinite Productfor1r,2803.12Integration of Rational Functions 282a. The Fundamental Types, 283 b. Integrationof the Fundamental Types, 284 c. PartialFractions, 286 d. Examples of Resolutioninto Partial Fractions. Method ofUndetermined Coefficients, 2883.13Integrationof SomeOtherClassesofFunctions 2908. Preliminary Remarks on the RationalRepresentation of the Circle and theHyperbola,290 b. Integration ofR(eos x,sin x), 193 c. Integration ofContents xvR(cosh x, sinh x), 294 d.Integration ofR(x, ~ ! 1 = - ~ ) , 294 e. Integration ofR(x, ~ = t ) , 295 f. Integration ofR(x, ~ ) , 295 g. Integration ofR(x, Vax2 + 2bx + c), 295 h. FurtherExamples of Reductionto Integrals of RationalFunctions, 296 i.Remarks onthe Examples,297Part C Further Steps in the Theoryof IntegralCalculus 2983.14Integralsof ElementaryFunctions 2988. Definition of Functions by Integrals. EllipticIntegrals andFunctions, 298 b. OnDifferentiation and Integration, 3003.15Extension of the Concept of Integral 3018. Introduction. Definition of "Improper"Integrals, 301 b. Functions with InfiniteDiscontinuities, 303 c. Interpretation asAreas,304 d. Tests for Convergence, 305e. Infinite Interval of Integration, 306 f. TheGamma Function, 308 g. The DirichletIntegral, 309 h. Substitution. FresnelIntegrals, 3103.16The Differential Equations of theTrigonometric Functions 3128. Introductory Remarks on DifferentialEquations, 312 b. Sin x and cos xdefinedbya Differential Equation andInitial Conditions,312PROBLEMS 314Chapter 4Applications in Physics and Geometry 3244.1 Theory of Plane Curves 3248. Parametric Representation, 324 b. Changeof Parameters, 326 c. Motion along a Curve.Time as the Parameter. Example of thexvi ContentsCycloid, 328 d. Classifications of Curves.Orientation, 333 e. Derivatives. Tangent andNormal, in Parametric Representation, 343f. The Length of a Curve, 348 g. The ArcLength as a Parameter, 352 h. Curvature,354 i. Change of Coordinate Axes.Invariance,360 j. Uniform Motion in theSpecial Theory of Relativity, 363 k.IntegralsExpressing Areawithin Closed Curves, 365I. Center of Mass andMoment of a Curve, 373m. Area and Volume of a Surface ofRevolution, 374 D. Moment of Inertia, 3754.2 Examples 3768. The Common Cycloid, 376 b. TheCatenary, 378 c. The Ellipse and theLemniscate, 3784.3 Vectors in Two Dimensions 379a. Definition of Vectors by Translation.Notations,380 b. Addition and Multiplicationof Vectors, 384 c. Variable Vectors, TheirDerivatives, and Integrals, 392 d. Applicationto Plane Curves. Direction, Speed, andAcceleration, 3944.4 Motion of a Particle under Given Forces 397a. Newton's Law of Motion, 397 b. Motionof Falling Bodies, 398 c. Motion of a ParticleConstrainedto a Given Curve, 4004.5 Free Fall of a Body Resisted by Air 4024.6 The Simplest Type of Elastic Vibration 4044.7 Motion on a Given Curve 405a. The Differential Equation and Its Solution,405 b. Particle Sliding down a Curve, 407c. Discussion of the Motion, 409 d. TheOrdinary Pendulum, 410 e. The CycloidalPendulum, 411Contents xvii4.8 Motion in a Gravitational Field 4138. Newton's Universal Law of Gravitation, 413b. CircularMotion about the Center ofAttraction, 415 c. Radial Motion-EscapeVelocity, 4164.9 Work and Energy 4188. WorkDone by Forces during aMotion, 4J8b.WorkandKinetic Energy. Conservation ofEnergy, 420 c. TheMutual Attraction ofTwoMasses,421 d. The Stretching of aSpring, 423 e. The Charging of a Condenser,423APPENDIX 424A.I Properties of the Evolute 424A.2 Areas Bounded by Closed Curves. Indices 430PROBLEMS 435Chapter 5 Taylor's Expansion5.1 Introduction: Power Series4404405.2 ExpansionoftheLogarithmandtheInverseTangent 442a. The Logarithm, 442 b. The InverseTangent, 4445.3 Taylor's Theorem 445a.Taylor's Representation of Polynomials, 445b. Taylor's Formula forNonpolynomialFunctions, 4465.4 ExpressionandEstimatesfor theRemainder 447a. Cauchy's andLagrange's Expressions, 447b. An Alternative Derivation of Taylor'sFormula, 4505.5 Expansions of the Elementary Functions 453a. The ExponentialFunction, 453xviii Contentsb. Expansion of sin x, cos x, sinh x, cosh x, 454c. The Binomial Series, 4565.6 Geometrical Applications 4578. Contact of Curves, 458 b. On the Theoryof Relative Maxima andMinima, 461APPENDIX I 462A.I.l ExampleofaFunctionWhichCannot BeExpanded in a Taylor Series 462A.I.2 Zeros and Infinites of Functions 463a. Zeros of Order n, 463 b. Infinity of Orderv,463A.I.3 Indeterminate Expressions 464A.I.4 TheConvergence of the Taylor Series of aFunction with Nonnegative Derivatives ofall Orders 467APPENDIX nINTERPOLATION 470A.II.l The Problem of Interpolation. Uniqueness 470A.II.2 Construction of the Solution. Newton'sInterpolation Formula 471A.II.3 The Estimate of the Remainder 474A.II.4 The Lagrange Interpolation Formula 476PROBLEMS 477Chapter 6 Numerical Methods 4816.1 Computation of Integrals 4828. Approximation by Rectangles, 482b. Refined Approximations-Simpson's Rule,483Contents xix6.2 Other Examples of Numerical Methods 4908. The "Calculus of Errors",490b. Calculation of7r, 492 c. Calculation ofLogarithms, 4936.3 Numerical Solution of Equations 494a. Newton'sMethod,495 b. The Rule of FalsePosition, 497 c. TheMethod of Iteration, 499d. Iterations andNewton's Procedure,502APPENDIXA.I Stirling's FormulaPROBLEMSChapter 7InfiniteSums and Products504504S075107.1 The Concepts of Convergence and Divergence 511a. BasicConcepts, 511 b. AbsoluteConvergence andConditional Convergence, 513c. Rearrangement of Terms,517d. Operations with Infinite Series, 5207.2 Tests for Absolute Convergence andDivergence 520a. The Comparison Test. Majorants, 520b. Convergence TestedbyComparisonwiththeGeometric Series, 521 c. ComparisonwithanIntegral, 5247.3 Sequences of Functions 526a. Limiting ProcesseswithFunctions andCurves, 5277.4 Uniform andNonuniformConvergence 529a. General Remarks andDefinitions, 529b. ATest forUniform Convergence, 534c. Continuity of the Sum of a UniformlyConvergent Series of Continuous Functions, 535d. Integration of UniformlyConvergentSeries, 536 e. Differentiation of InfiniteSeries, 538xx Contents7.5 Power Series 5408. Convergence Properties of Power Series-Interval of Convergence, S40 b. IntegrationandDifferentiation of Power Series, 542c. Operations withPower Series, 543d. Uniqueness of Expansion, 544 e. AnalyticFunctions, 5457.6 Expansion of Given Functions in Power Series.Method of Undetermined Coefficients.Examples 5468. The Exponential Function, 546 b. TheBinomialSeries, 546 c.The Series forarcsinx, 549 d. The Series forar sinhx = log[x + v(l +x 2)],549e. Example of Multiplication of Series, 550f. Example of Term-by-Term Integration(Elliptic Integral), 5507.7 Power Series with Complex Terms 5518. Introduction of Complex Terms into PowerSeries. Complex Representations of theTrigonometric Function, SSt b. A Glance atthe General Theory of Functions of aComplexVariable, 553APPENDIX 555A.I MultiplicationandDivisionof Series 5558. Multiplication of Absolutely ConvergentSeries, 555 b. Multiplication andDivision ofPower Series, 556A.2 Infinite Series and Improper Integrals 557A.3 Infinite Products 559A.4 Series Involving Bernoulli Numbers 562PROBLEMS S64Chapter 8"TrigonometricSeriesContents xxi5718.1 Periodic Functions 5728. GeneralRemarks. Periodic Extension of aFunction,572 b.Integrals Over aPeriod, 573c. Harmonic Vibrations, 5748.2 Superposition of Harmonic Vibrations 576a. Harmonics. TrigonometricPolynomials, 576b.Beats, 5778.3 Complex Notation 582a. GeneralRemarks, 582 b. ApplicationtoAlternating Currents, 583 c. ComplexNotation for Trigonometrical Polynomials,585d. A Trigonometric Formula, 5868.4 Fourier Series 5878. Fourier Coefficients, 587 b.BasicLemma,588tOO sin z 7rc. Proof of -- dz = -, 589o z 2d. Fourier Expansion fortheFunctioncP (x) =x,591 e. TheMainTheoremon Fourier Expansion, 5938.5 Examples of Fourier Series 598a. PreliminaryRemarks, 598 b. Expansion ofthe FunctioncP (x) =x2, 598 c.Expansionof xcos x, 598 d. TheFunction/(x) =Ixl,600 e.APiecewiseConstant Function, 600 f. The Functionsinlxi,601 g. Expansion of cosJ.LX.Resolution of the Cotangent into PartialFractions. The Infinite Product fortheSine, 602 h. Further Examples, 6038.6 Further Discussion of Convergence 604a. Results, 604 b. Bessel's Inequality,604xxii Contentsc. Proof of Corollaries (a), (b), and (c), 60Sd. Order of Magnitude of the FourierCoefficients Differentiation of FourierSeries, 6078.7 Approximation by Trigonometric and RationalPolynomials 608a.GeneralRemarkonRepresentations ofFunctions, 608 b. WeierstrassApproximationTheorem, 608 c. Fejers TrigonometricApproximation of Fourier PolynomialsbyArithmetical Means, 610 d. ApproximationintheMean andParseval'sRelation, 612APPENDIXI 614A.I.t Stretching of the Period Interval. Fourier'sIntegral Theorem 6t4A.I.2 Gibb's Phenomenon at Points ofDiscontinuity 616A.I.3 Integration of Fourier Series 618APPENDIXII 619A.II.l Bernoulli Polynomials and TheirApplications 619a.Definition andFourier Expansion, 619b. Generating Functions andthe Taylor Seriesof the Trigonometric andHyperbolicCotangent,621 c. The Euler-MaclaurinSummationFormula, 624 d. Applications.Asymptotic Expressions, 626 e.Sums ofPower RecursionFormula for BernoulliNumbers, 628 f'. Euler's Constant andStirling's Series, 629PROBLEMS 6YContents xxiiiChapter 9Differential Equationsfor the SimplestTypes of Vibration 6339.1 Vibration Problems of Mechanics and Physics 6348. The Simplest Mechanical Vibrations, 634b. Electrical Oscillations, 6359.2 Solution of the Homogeneous Equation.Oscillationsa.The FomalSolution, 636 b. PhysicalInterpretation of the Solution, 638c. Fulfilment of GivenInitial Conditions.Uniqueness of the Solution, 639Free6369.3 The Nonhomogeneous Equation. ForcedOscillations 6408. General Remarks. Superposition, 640b. Solution of the NonhomogeneousEquation, 642 c. TheResonance Curve, 643d. Further Discussion of the Oscillation, 646e. Remarksonthe Constructionof RecordingInstruments, 647List of Biographical DatesIndex6506531IntroductionSinceantiquitytheintuitive notions ofcontinuous change, growth,and motion, have challenged scientific minds. Yet, the way to theunderstandingof continuous variationwasopenedonlyintheseven-teenthcenturywhenmodernscience emergedandrapidlydevelopedincloseconjunctionwithintegral anddifferential calculus, brieflycalledcalculus, andmathematical analysis.The basic notions of Calculus are derivative and integral: thederivativeis ameasurefor therateof change, theintegral a measureforthe totaleffect of aprocess of continuous change. A preciseunder-standingofthese concepts and their overwhelmingfruitfulness restsupontheconcepts of limit andof functionwhichin turndependuponan understandingofthecontinuumofnumbers. Onlygradually, bypenetratingmoreand more into the substance ofCalculus, can oneappreciateitspower andbeauty. Inthisintroductory chapter weshallexplain the basic concepts of number, function, and limit, at firstsimply andintuitively, andthenwithcareful argument.1.1 The Continuumof NumbersThe positive integers or natural numbers 1, 2, 3, . .. are abstractsymbols for indicating "how many"objectsthereareinacollection orset of discreteelements.Thesesymbolsarestrippedof all referencetotheconcretequalitiesofthe objects counted, whether they are persons, atoms, houses, oranyobjectswhatever.The natural numbers are the adequate instrument for countingelements ofa collectionor "set." However, theydonot suffice foranother equally important objective: to measure quantities suchasthelength of acurveandthevolumeorweight of abody. Thequestion,12 Introduction Ch. 1"how much ?", cannot be answered immediately in terms of the naturalnumbers. The profound need for expressing measures of quantitiesinterms of whatwewouldliketocallnumbersforcesusto extendthenumber concept sothat we maydescribe a continuous gradationofmeasures. This extensionis called the number continuum or the systemof"real numbers" (a nondescriptive but generally accepted name).Theextensionof thenumber concept tothat of thecontinuumis soconvincinglynatural that it wasusedbyall thegreat mathematiciansand scientists of earlier timeswithout probing questions. Not until thenineteenthcenturydidmathematiciansfeel compelledtoseekafirmerlogical foundationfor thereal number system. Theensuingpreciseformulationof theconcepts, inturn, ledtofurther progressinmathe-matics. We shall beginwith anunencumbered intuitive approach, andlater on weshall give adeeper analysis of thesystem of realnumbers.1a. TheSystem of Natural Numbers and ItsExtension. Counting and MeasuringTheNatural and theRational Numbers. Thesequenceof "natural"numbers1,2, 3, ...is consideredasgiventous. Weneednot discusshowtheseabstract entities, thenumbers, maybecategorizedfromaphilosophical point of view. For themathematician, andforanybodyworking withnumbers, it is important merely to know the rules or lawsby which they may be combined to yield other natural numbers. Theselawsformthe basis ofthefamiliar rules for addingand multiplyingnumbers in the decimal system; they include the commutative lawsa +b= b + a and ab=ba, the associative laws a + (b +c) =(a +b) +c and a(be) =(ab)c, the distributive law a(b + c) =ab + ac,thecancellationlawthat a + c =b + c implies a =b, etc.The inverse operations, subtraction and division, are not alwayspossible within the set of natural numbers; we cannot subtract 2from 1or divide 1by 2 and stay within that set. Tomake theseoperations possible without restriction we are forced to extend theconcept of number by inventing thenumber 0, the"negative" integers,and the fractions. The totality of all these numbers is called the class orset of rational numbers; theyareall obtainedfromunitybyusingthe"rational operations" ofcalculation, namely, addition, subtraction,multiplication, anddivision.2Arational number canalways bewrittenintheformplq, where p1 Amorecompleteexpositionisgivenin What Is Mathematics?byCourant andRobbins, Oxford University Press, 1962.Z The word "rational" here does not mean reasonable or logical but is derived fromthe word "ratio" meaning the relative proportion of two magnitudes.Sec. J.JThe Continuumof Numbers 3if xispositive orzero,and q areintegers and q7':- O. We canmake thisrepresentationuniqueby requiring that q ispositive andthat p and q haveno common factorlargerthan I.Withinthedomainof rational numbers all therational operations,addition, multiplication, subtraction, anddivision(except divisionbyzero), canbeperformedandproduceagainrational numbers. As weknowfromelementaryarithmetic, operations withrational numbersobey the same laws as operations with natural numbers: thus therational numbers extend the systemof positive integers in a com-pletely straightforwardway.Graphical RepresentationofRational Numbers. Rational numbersare usually represented graphically by points on a straight line L,the nunlber axis. Taking anarbitrary point of Las the origin or point 0/ \----------+-1-+------11----- Lo pr-----+--xI \------+--\----\I---+---------LP 0~Figure 1.1 Thenumber axis.and anotherarbitrary point as the point1, we use thedistancebetweenthese two points to serve as a scale or unit of measurement and define thedirectionfrom0to I as upositive." The line witha direction thusimposediscalledadirectedline. It iscustomarytodepict Lsothatthe point] is tothe right of the point 0 (Fig. 1.1). The location of anypointPonLiscompletelydeterminedbytwopiecesof information:the distance of P fromthe origin 0 and the direction from0 toP (to theright or left of 0). Thepoint PonLrepresentingapositiverationalnumber liesatdistancexunitstotherightofO. Anegativerationalnumber x is represented by the point -x units to the left of O. Ineithercasethedistancefrom0tothepoint whichrepresents xiscalledtheabsolutevalue of x, written lxi,andwehaveIxl= { x,-x, if xisnegative.We notethatIxl isnevernegative andequals zeroonlywhen x =o.4 Introduction Ch. 1Fromelementary geometrywerecall thatwithrulerandcompassitis possible to construct a subdivision of the unit length into any numberof equalparts. Itfollowsthatanyrational length canbeconstructedand hence that the point representing a rational number x can befoundby purely geometrical methods.In this way we obtain a geometrical representation of rationalnumbers by points on L, the rational points. Consistent with ournotation for the points 0 and1, we take the liberty of denoting both therational number and the corresponding point on L by the same symbol x.Therelationx < y for tworational numbers means geometricallythat the point x lies to the left of the point y. In that case the distancebetween the points is y- xunits. If x > y, the distance is x- y units.Ineither casethe distance betweentwo rational points x, y of L isIY - xl units andis againarationalnumber.p------Lo .! 1. p p+1q q q -q-Figure 1.2AsegmentonLwithendpointsa, bwherea O. Consequently, twoinequalities a > bandc > d canbe added to yieldthe inequalitya + c > b + dsince(a + c)- (b + d) =(a- b) + (c- d)is positive as the sumof two positive numbers. (Subtracting theinequalities to obtain a- c > b - dis not legitimate. Why?) Aninequality canbemultiplied by0 positil1e number; that is, if a > bandc > 0, thenac > be. For theproof, we observe thatoc- be =(a- b)eispositive sinceit istheproduct of positivenumbers. If cisnegative,we canconcludefroma > b thatac < be. Moregenerally, itfollowsfroma > b > 0 andc > d > 0 thatac > bd.It is geometricallyobvious that inequalityis transitive; that is, ifa > bandb >c, thena >c. Transitivityl alsofollows immediatelyfromthepositivity of thesum(a- b) + (b- e) =a-c.The preceding rules also hold if we replace the sign>by~ everywhere.Letaandb be positivenumbersandobserve thata2- b2= (a +b)(a- b).Since a + b is positive, weconcludethat 0 2> b2followsfroma > b.Thus an inequality between positive numbers can be "squared."Similarly, a2:2b2whenever a:2b ~ O. Fromthe equation1 2 2a - b = --(a - b ),a +bvalidfor all positiveaandb, it followsthat theconverseisalsotrue;that is, for positive a and b, a2> b2implies a > b. Applying thisresult to the numbers a= J-;, b = Jy, for arbitrary positive realnumbers x, y, we find2that J; > Jywhenx > y. Moregenerally,J-;:2 Jy whenever x ZY :2O. Hence it is legitimate to take the1 Transitivity justifies theuse of the compound formula "0 < b < c ..."to express"0 < bandb < c, etc." Avoidnontransitivearrangementslike x z; theseare confusing andmisleading.2 Hereandhereafter thesymbol vi; for z ~ 0denotes that nonnegativenumberwhose squareisz. Withthis conventionlei=V ~ foranyrealc sincelei ~ 0 and!c1 2=c2 From thisweobtainthe importantidentity Ixyl = Ixl IylsinceIxyl2= (xy)2= X2y 2 = (Ixl lyI)2.14 Introduction Ch. 1squareroot ofboth sides ofaninequalitybetweennonnegative realnumbers.Suppose that a and b arepositive and n is apositive integer. Inthefactorizationan- bn= (a- b)(an-l+ an-2b + ... +bn-l)the second factor is positive. Thus an- bnhas the same sign as a - b;if an > bn, then a > b and if an < bn, then a < b.Most inequalitiesweshallencounteroccurintheformof estimatesfortheabsolutevalue of anumber. Werecall that Ixl isdefinedtobex for x~ 0 and -xfor x < O. Wemayalsosay that Ixl isthelargerof the twonumbers xand -x when xisnot zeroandis equal tobothof them when x is zero. The inequality Ixl Sa then states that neitherxnor -x exceeds a, that is, that x Sa and -x Sa. Since -x Sa isequivalent tox ~ -a,wesee that the inequalityIxl Sa meansthat x------+-----+-------+-------LY"(l-a Xl) Xo +aFigure 1.5 The interval Ix - xol ~ a.lies in the closedinterval -a~ x ~ awithcenter 0andlength2a.The inequalityIx- xol ~ a thenstates that -a~ x- xo ~ a or thatXo - a S x SXo+a, thus, that x lies in the closed interval with center Xoandlength 2a(see Fig. 1.5). Similarly, the -neighborhood(xo- ,Xo+ E) of apointxo, thatis, theopeninterval Xo - (x) =l/x and g(u)=sin u.It is useful to interpret the compound functions in termsof mappings.The mapping> takes everypoint xof the interval [a, b]intoa point uinthe interval [at, {3]; themapping g takesanyvalue u in[at, {3] into apoint y. The mapping/isthe "symbolic product"geP of the mappings1 This is done on p. 152.Sec. J.3 The Elementary Functions 53g and4>,that is, themapping carrying out 4> and g successively, in thatorder; for any xin[a,b]we formitsmap u under themappingep, andthenapply gtotheimageu= 4>(x) , obtaining g(eI>(x)) = [(x) = y(seeFig. 1.39). Such a symbolic product g4> is natural and meaningfulfor any type of operation~ it signifies that we first performe/>, andthen, onthe result, perform g.l We must not confuse the symbolicproduct gel>= g e e / ~ of two functions with the ordinary algebraic productg(x) .e/>(x) ofthe functions, in which bothg(x) and ep(x) are formedfor the same argument x (the mappings applied to the same point)andthe product o[ thevalues of thefunctionsisformed.Naturally,symbolic products cannot be expectedtobe commutative.In general,K(e/ and cp(g) are not the same, evenwhere both are defined;yxFigure1.39 Symbolic product g>= f of twomappings.the order inwhichoperationsare performedmatters verymuch. If,for example, ep stands for the operationof"adding 1 toa number"and gfortheoperationof "multiplying anumberby2,"theng(cp(x)) = 2(x +1)= 2x + 2, cp(g(x = (2x) +I =2x +1.(SeeFig. ] .40.)In order to be able to form the symbolic productgcp of two mappings,theHfactors" g andr/> must fittogetherinthesensethatthe domain ofgmust includethe rangeofcp; thus wecannot formgr/> wheng(u) = . J ~ , and cp(x) = -1- x21 Thattheproductg ~ correspondsto firstcarrying out ~ and then g(inthat order)seems unnatural at first glance, but actuallycorrespondstotheconventionalwaysadoptedinmathematics of writingthe argumentxof afunction I(x)tothe right ofthesymbol I forthefunction. Thus, forexample, insin (log x) it isalwaysunder-stood that we first formthe logarithm of x and then take the sine of that, and not theother way around.S4 IntroductionCh. JItisusefultoconsiderfunctionswhichare compoundedmorethanonce. Suchafunctionisf(x) =I +tan (x2),whichcanbebuilt upby successive compositions4>(x) = x2, 1p(4))= 1 +tang(1p) ==f(x).Wewouldwritesymbolicallyf=2x + 1 2(x + 1)ooogxFigure1.40 Noncommutativity of mappings.Inverse FunctionsThenotionof "inversefunction" becomesclearerinthecontext ofproduct of mappings. Consider the mapping ep associating with apoint xofthe domain of J the image u = J(x). Assume that ourmappingepissuchthatdifferent xarealwaysmappedintodifferent u.The mapping is thencalled "one toone." Then avalueu is the imageof at most one value x. We can associate with every u in the range ofthe value x = g(u) of which uis the image under the mapping cP.Inthisway wehave defined amapping g whose domainis therange of and which when applied to an image u =cP(x) ofthe ep-mappingreproduces the original value x, that is, g(c/>(x) =x. We call gtheinverseof C/>. It is characterized bythe symbolicequationgc/>x =x.The IdentityAJappingWedefinetheidentitymappingIastheonethat mapseveryxintoitself; for the inverse g of cP then,gc/>= /.1The mapping / plays the samerole for symbolic multiplication as the number 1 in ordinary multiplica-tion; multiplication byI does not changea mapping. Accordingly,the equationgc/> =I suggests the notation g =4>-1 for the inverse of 4>.For example, the inverse x= arc sin u of the function u =sin x isoftendenotedbyx= sin-1u.21More precisely gcp agrees with I, in the domain of cp.2 Thismust not be confusedwiththereciprocal 1j(sin u).Sec. 1.4Sequences 55From thedefinition of theinverse g ofcP it followsimmediately thatalsocP is the irrverse of gso that not only g(cP) = x but also c/>(g(u)) =u.A monotone functionu=ep(x) defined in aninterval a ~ x ~ bclearlydefines a1-1mapping of that interval. If, in addition, ep is continuous, thenaswe saw earlier asaconsequence of the intermediate value theorem (p. 44),the range of rp is the interval with end pointsep(a) andep(b). In that case theinverse gof 4> existsandisagainmonotoneandcontinuousinthat latterinterval. Asamatterof fact themonotonecontinuous functionsaretheonlycontinuous,functions that have inverses or define one-to-one mappings. Indeed,let u= ep(x) beacontinuousfunctionintheclosedinterval [a, b]mappingdifferent x of the interval intodifferent u. Theninparticular the valuesrp(a) =(X and4>(b) = fJaredistinct. We assume, say, that (X (a), andthis alsocontradicts the 1-1nature ofep.Animportant, almost obvious propertyof compoundfunctions, isthatg(4)(x)) = I(x) is continuous (where defined) ifg and4> are. Indeed,forgivenpositive wehaveasaconsequence of thecontinuityof thefunction g. Since, however,eP is alsocontinuous, we certainlyhave Ic/>(x) - 4>(xo) I I, we put ex = 1 + h, wherehis positive, andat onceseefromour inequality that as n increases exndoes not tendto any definitelimit, but increases beyondall bounds. We say that exntends toinfinityasn increasesor that lI.71becomes infinite; insymbols,limexn= 00 (ex >1).Weexplicitlyemphasizethat thesymbol 00 doesnot denoteanumberand that wecannot calculatewithit accordingtotheusual rules; state-mentsassertingthat aquantityis or becomesinfinitenever havethesame senseas an assertioninvolvingdefinite quantities. Inspiteofthis, such modes of expression and the use of the symbol 00 areextremely convenient, aswe shalloftensee in the followingpages.If ex = -I,thevalueof exndoes not tend to any limit,but as n runsthroughthesequenceof positiveintegers exntakesthevalues +1 and-1 alternately. Similarly, if ex < -I the value of exnincreasesnumericallybeyondall bounds, but itssignisalternatelypositiveandnegative.f. Geometrical Illustration of theLimits of ex 11and '\IpIf we consider the graphs of the functions y =xnand y= x1/n=\ y ~and restrict ourselves for the sake of convenience to nonnegative valuesof x, thepreceding limitsare illustratedbyFigs. 1.44 and1.45 respec-tively. Wesee that inthe interval from 0 to1 the curvesy =xncome66 IntroductionyI...,IiiI"I IIIIII1:I: IIIIII:I'E!

Figure 1.44 xnasn increases.Ch. 1closer and closer to the x-axis as nincreases, whereas outside thatintervalthey climb more andmore steeply andapproachalineparalleltothey-axis. All thecurvespassthroughthepoint withcoordinatesx =],y =1 andtheorigin.The graphs of the functions ]I =x1/ n= +Vi;, come closer andclosertothelineparallel tothex-axis andat adistance 1aboveit;againall thecurvesmust passthroughtheoriginandthepoint (l,1).Henceinthelimit thecurves approachthebrokenlineconsistingofthepart of they-axisbetweenthepointsy =0andy =1 andof theparallel tothex-axisy =1. Moreover, itisclear that thetwofiguresare closely related, as one wouldexpect from the fact that the functionsy=aretheinversefunctions ofthenthpowers, fromwhichweinferthat for eachnthegraphof y =xnistransformedintothat ofy =\Y"X byreflectionin the line y =x.Sec. 1.6yThe Limit of aSequence 67- - - f - . : - - - - - - - - - - - - - - - - - - - - - - ~ xFigure1.45 XI/'1as 11 increases.g. TheGeometric SeriesAn example of a limit familiar fromelementary mathematics ISfurnishedbythe geometric series1 + q +q2 + ... +qrt 1 = Sn;thenumber qiscalledthecommonratioor quotientof theseries. Thevalue of thissummay, asiswell known, beexpressedintheform1 _qnS =--n l-qprovidedthat q 1; we canderivethisexpressionbymultiplying thesumSn byqand subtracting the equation thus obtained fromtheoriginalequationor wemayverifytheformulabydivision.What becomes ofthe sumSnwhennincreases indefinitely? Theanswer is: The sequence ofsums Sn has a definite limit S if qlies68 Introductionbetween -1and +1, these endvaluesbeing excluded,andS =limS =_1_.n 1 _qCh. 1In order to verify this statement we write Sn as (1 - qn)/(1- q)= 1/(1- q) - qn/(l - q). We have already shown that providedIql1 thath2 < _2_.n-n_1'henceWenowhaveJ2hn IrxnCh. 1Formally, the limit of the an is of the indeterminate type00/00 alreadyencountered in Examplec. We assert that in this example the sequenceof numbersan= n/rxntendstothe limit zero.Fortheproof weput rx = 1 + h, where h > 0, andagainmakeuseof the inequalityHencefor n >1n 2a =---< .n (1 + h)n (n- 1)h2Sinceanispositiveandtheright-handsideof thisinequalitytendstozero, anmust alsotendtozero.1.7 Discussionofthe Concept ofLimitQ. Definition of Convergence and DivergenceFromthe examples discussed in Section1.6 we abstract the followinggeneral concept of limit:Suppose that for a given infinite sequence ofpoints aI'a2,a3,thereis a number I such that every open interval, no matter how small, markedoff about thepoint I, containsall thepoints anexcept for at most afinite number. The number I is then calledthe limit of thesequenceaI' a2,, or we say that the sequence a1'Q2' is convergent andconverges to I; in symbols, liman= I.n-+C()The following definition of limit is equivalent:To anypositive number f, no matter howsmall, we can assign asufficientlylargeintegerN= N( f)suchthat fromtheindexNonward[that is, forn > N( f)] we always haveIan- /1 N( E). If wewriteab- anbn = b(a - an) + an(b- bn)and recall that there is a positive bound M, independent of n, such thatlanl< M,weobtainSince the quantity (Ihl + M)E can be made arbitrarily small by choosingEsmall enough, thedifferencebetweenabandanbn actuallybecomesas small as we please for all sufficiently large values ofn; this isprecisely thestatement made in theequationab=lim anbn.n-+ 00Sec. 1.7Discussionof tire Concept of Limit 73Usingthis exampleas amodel, thereader canprovetherules fortheremaining l'ationaloperations.Bymeansof theserules manylimitscanbeevaluated easily; thus,wehave1 - l.-1. n2- 1. n21m 2 = hm =1,n-+00 n + n + 1 n.....00 1 +1 1-+-n n2sinceinthesecondexpressionwecanpassdirectlytothelimit inthenumerator anddenominator.The following simple rule is frequently useful: If lim an= aandlimbrz = b,and if in additionan > bnfor every n, then a~ b. We are,however, bynomeansentitledtoexpectthatawill alwaysbe greaterthanb, as is shownbythesequencesan =lin, bn =1/2n, for whicha =b = O.c. IntrinsicConvergenceTests. MonotoneSequencesInalltheexamples giventhe limit of thesequence consideredwasaknownnumber. In fact, toapplythe abovedefinition of limit of ase-q uence we must know the limit before we can verify convergence. If theconcept of limit of a sequence yielded nothing more than the recognitionthat someknownnumberscanbeapproximatedbycertainsequencesofother knownnumbers, we shouldhavegainedverylittlefromit.The advantage of the concept of limit in analysis lies essentially on thefact that important problems often have numerical solutions which maynot otherwisebedirectlyknownorexpressible, but canbedescribedas limits. The whole of higher analysis consists ofa succession ofexamples of this fact which will become steadily clearer in the followingchapters. The representation ofthe irrational numbers as limits ofrational numbers may be regardedas the first and typical example.Any convergent sequence of known numbers at, Q2' defines anumber I, its limit. However, theonly test forconvergence that arisesfrom the definition of convergence consists in estimating the differencesIan - 1/, andthisisapplicableonlyif thenumber 1isknownalready.It is essential to have "intrinsic" tests for convergence that do notrequire an a priori knowledge of the value of the limit but only involvethetermsof thesequencethemselves. Thesimplest suchtest appliestoaspecial classof sequences, themonotonesequences, andincludesmost of the important examples.74 Introduction Ch. JLimitsof Monotone SequencesAsequence aI' a2' ...iscalled monotonically increasing if eachterman is larger,or at least not smaller thanthepreceding one; thatis,Similarly, the sequence is monotonically decreasing if an~ an-l forall n. A monotone sequence is one that is eithermonotonicallyincreas-ingor decreasing. Withthis definitionwehavethebasic principle:A sequence that is both monotone and bounded converges.1This principle is convincingly suggested, but not proved, by intuition;it is intimatelyrelatedtothepropertiesof real numbersandinfact isequivalenttothe continuity axiomforreal numbers.Theaxiom(seeSection 1b) thateverynestedsequenceof intervalscontains a point is easilyseentobeaconsequenceof theconvergenceof bounded monotone sequences. For let [a}, bI ], [a2, b2], be asequence of nested intervals. By the definition of nested sequences wehaveObviously, the infinite sequence aI'a2, ... is monotonically increasing.It is also bounded since a1~ an~ hI for all n. Hence I =lim anexists. Moreover, foranymandforanynumbern > mwehaveam~ an~ bm HencealsoThus all intervals ofthe nestedsequencecontain one andthe samepoint I. (Thattheyhavenootherpointincommonfollowsfromthefurther propertylim (bn- an) =0 of nestedsequencesof intervals.)Cauchy's Criteria for ConvergenceAconvergent sequence is automaticallyboundedbut need not bemonotone (see Example b, p. 62). Hence, in dealing with generalsequences, it is desirable to havea test for convergencethat is also1 The assumptionof boundedness is essential sincenounboundedsequencecanconverge. Oberve that a monotonically increasingsequence 017 02, .. is always"bounded frombelow": an ~ Ql foraLi n. Inorder toprove that a monotonicallyincreasingsequenceconverges it is sufficient thentofinda number Msuchthatan~ Mforall n.Sec. 1.7 Discussionof the Concept of Limit 75applicable to nonmonotone sequences. This need is satisfied by asimple condi-tion, the Cauchy test for convergence; this criterioncharacterizes sequences of real numbers which have a limit; mostimportantlyitdoesnot requirea prioriknowledgeof thevalueof thelimit: Necessary and sufficient forconvergence of a sequence aI' a2, is that the elements anof thesequence withsufficientlylarge indexndiffer arbitrarily little from each other. Formulated precisely: asequence an is convergent if for every E > 0there exists a naturalnumber N= N(E) such that Ian- ami< Ewhenever n > Nand m >N.Geometrically, theCauchyconditionstatesthatasequenceconvergesif thereexist arbitrarilysmall intervalsoutsideof whichtherelieonlya finitenumber of points of the sequence. The correctness of Cauchy'stest forconvergence will beproved and its significance discussedin theSupplement.d. Infinite Series and theSummationSymbolAsequenceis justanorderedinfinitearrayof numbersa1,a2,An infinite seriesrequiresthetermstobeaddedintheorder inwhichtheyappear. Toarrive at aprecise meaning of the sumof aninfinite series weconsiderthe11th partial sumthatis, thesumof thefirst ntermsof theseriesThepartial sumsSn fordifferent n formasequenceandsoon. Thesum s of theinfinite series isthendefinedass= limSmn-+00providedthislimit exists. Inthat casewecall theinfiniteseriescon-vergent. If the sequenceSn diverges, the infinite series is called divergent;For example, the sequence 1, q, q2, q3, ... gives rise to the infinitegeometric series1 + q +q2 +q3 + ...whose partialsumsaresn =1 +q +q2 +... +qn-l.76 IntroductionForIql < 1 the sequence sn converges toward the limitChI I1S=--,l-qwhichthenrepresentsthesumof theinfiniteseries. For Iql ~ 1thepartial sums Sn have nolimit andthe series diverges (seep. 67).It is customary touse fora1 + a2 + ... + an thesymboln~ a kk=1whichindicatesthatthesumof theakistobetakenwithk runningthrough the integers from k= 1 tok= n. For example,4 1 1 1 1 11 ~ 1 k!stands for i! + 2! + 3! + 4! 'whereasf akb2kstands for a1b2+ a2b4+ a3b6+ ... + anb2nk=1More generally, i Ok means the sum of allOkobtainedby giving k thek=mvalues m, m +1, m +2, ... ,n. ThusIntheseexampleswehaveusedtheletter k for theindexof sum-mation. Of course, thesumisindependent of theletterdenotingthisindex. Thusn nSn= Zak= Zai k=l i=lWeusethe symbolC/)todenotethesumof thewholeinfiniteseries. Similarly, L Okwouldk=Ostand forthe sum of the infinite series00 + a1 +Q2 + ... , whose nthpartialsum isSn= Qo +0 1 + a2 + .. + an-I'Manyofour earlier results canbe writtenmore conciselyin thissummationnotation. Theformulaof p. 58, for thesumofthefirstn squares becomesik2= n(n +1)(2n +1) .k-l 6Sec.1.7Discussionof the Concept of Limit 77forIql 1 + 1 +~ 1 - - + ... + - 1 - - ... 1 - -- .2! m n! m m]f we now keep n fixedand let m increase beyond all bounds, we obtainontheleft thenumber Tandontheright theexpressionSmsothatT~ Sn' Thus T ZSn~ Tn for every value of n. We nowlet nincrease, sothat TntendstoT; fromthedoubleinequalityitfollowsthat T= lim Sn= e. Thiswas thestatement tobeproved.n-+CXlWe shall later (Section 2.6, p. 149) be led to this number e again fromstill another point of view.f. TheNumber7T as a LimitAlimitingprocesswhichinessencegoesbacktoclassical antiquity(Archimedes) is that by which the number1T is defined. Geometrically,1Tmeansthearea of thecircleof radiusone. Weregardit asobviousthat this areacanbeexpressedbya (rational or irrational) number,denotedby1T. However, thisdefinition is not of much help tous if wewishtocalculate the number withanyaccuracy. We then have nochoice but torepresent the number bymeans ofa limitingprocess,namely, as the limit of a sequence of knownand easily calculatednumbers. Archimedes already used this process in his method ofexhaustion, which consists ofapproximatingthe circle by means ofregular polygons with anincreasing number of sides fitting it more andmore closely. Ifwe letlm denote the area of the regular m-gon (polygonof msides)inscribedinthecircle, theareaof theinscribed2m-gooisgiven by the formula [proved by elementary geometry or fromtheexpression In = (nI2)sin(21Tjn)(seeFig. 1.46)]f2m= ~ J 2 - 2,,}1- (2f".fm)2,We now let m range, not through the sequence of all positive integers butthrough thesequence of powers of 2, that is, m= 2n; inother words,we form those regular polygons whose vertices are obtained by repeatedSec. 1.7Discussionof the Concept of Limit 81Figure 1.46bisection of the circumference. It is clear fromthe geometricinterpre-tationthat tlle Anformanincreasing andboundedsequence andthushave alimit which is the areaof the circle:7T = lim 12".n-+00This representation of 7T as a limit serves actually as a basis fornumerical computations; for, startingwiththe value f4 = 2, we cancalculatein order the terms ofoursequencetendingto 7T. Anestimateof theaccuracy with whichany termf2fi represents 7T canbeobtained byconstructing the lines touching thecircle and parallel tothesidesof theinscribed2ft-gon. These lines formacircumscribed polygonsimilar totheinscribed 2tl-gon and having largerdimensions intheratio1:cos (7Tj21l).Hence the area F2n of the circum-scribedpolygonmaybefound fromtheratiogivenbyf2n( 7T)2- = cos- .F2n 2r1Sincetheareaof thecircumscribedpolygonisgreater than that of thecircle, wehaveFor example,/8 = 2)"2,so that wehave the estimate- 4)22J2anhavethe limit I).Statementsabout limitscanbeexpressedinterms ofrational nul/-sequences, that is, sequences01' a2,of rational numbers for whichliman= O.n-+ooOne says an "becomes arbitrarily small as n tendsto infinity,"meaningthatfor anypositiverational , nomatter howsmall, the inequalitylanl fJn ~ y.d. Completeness of theNumberContinuum. Compactnessof Closed Intervals. Convergence CriteriaReal numbers make possible limit operations with rational numbers,buttheywouldbe of littlevalueif thecorrespondinglimitoperationscarriedout with themnecessitatedthe introductionofsome furtherkindof "unreal"numbers whichwouldhavetobefittedinbetweenthereal ones, andsoonadinfiPlitum. Fortunately, thedefinitionofreal number is so comprehensive that no further extensionof theSec. S.lLimitsandthe Number Concept 95number systemis possible without discarding one of its essentialproperties..(as "order"mustbe discardedforcomplexnumbers).PrincipleofContinuityThis completeness ofthe real number continuumis expressed bythe basic continuity principle (cf. p. 8): Every nested sequence ofintervalswithreal endpointscontainsareal number. Toprovethis,consider closed intervals [xn' Yn], each interval contained in the precedingone, whoselengthsYl1 - Xnformanull-sequence. We claim there is areal x containedin all [xmYn]: The sequencesXn and Ynwill then havexaslimit. Toprovethiswereplacethenestedsequence[xmYn] byanestedsequence ofrational intervals [am bn], containingthe [xmYnl.This rational sequence willthendefine the desired real numberx. Foreach11 let 011bethelargest rational number of theform pl2t'less thanXmandbnthesmallestrational numberof theformql2UgreaterthanYmwhere pandq areintegers. Clearly, theintervals [am bnl formanestedsequencerepresentingareal numberx. If xlayoutsideoneoftheintervals [xm, Ym], sayx 11. It follows that everyinterval containing X contains almost all Xn, or X is thelimit ofthesequence.Cauchy's Convergence CriterionThe condition that a sequence is bounded and monotone is sufficientfor convergence. The significance of this statement is that it oftenpermits us to prove existence of the limit of a sequence withoutrequiringa priori knowledgeofthe value ofthelimit; in addition,boundedness and monotonicityofa sequence are properties usuallyeasy to check in concrete applications. However, not every convergentsequenceneedbe monotone(althoughit hastobebounded)andit isimportant to have a more generally applicable criterion for convergence.Sec. S.lLimitsandtheNumber Concept 97Suchis the intrinsicconvergencetest of Cauchywhichis a necessaryandsufficient conditionfor theexistenceofthelimit ofa sequence.The sequence Xl' X2, Xa, ...converges if and only iffor every positivethere exists anN such that IXn- xml 0, showthat thedomainoff contains an open interval aboutawhere f(x) >O.2. In the definition of continuity show that the centered intervalsIf(x) - f(xo)I 0,show that anneed not converge.16. Prove therelationforanynonnegativeinteger k. (Hint: Useinductionwithrespect tok andusetherelationi[ik+1- (i - 1)":+1] = nk-+1,i"-lexpanding (i - 1)k+l inpowers of i.)SECTION1.7, page 70*1. Let 01andb1beanytwopositive numbers, andlet 01 < b1 Let 02andb2be definedbythe equationsSimilarly,letand, in general,/---an=v an-1bn-1,ba= 2 + b22 'b= an-1 + bn-1n 2 .Prove (a) that the sequence aI' 02' ... , converges, (b) that the sequenceb1, b2,, converges, and(c) that thetwosequences havethesamelimit.(Thislimit is calJed the aritillnetic-geonletric 1nean of a1and bl .)114 Introduction*2. Prove that the limit of the sequenceY2, J2 + v2, J2 + J2 +Y2, ...(a) exists and(b)itis equal to2.*3. Prove that the limit of the sequence1 1 1On =- +-- + ... +-n n + 1 2nCh. 1exists. Show that the limit is less than1 but not lessthan i.4. Prove that the limit of the sequenceb =_l_+ ... + ~n n+1 2nexists, isequal tothelimit of thepreviousexample.S. Obtain the following bounds for the limit L in the two previousexamples: 37/60 b.The Generalized Mean Value Theorem. Insteadof thesimplearithmeticaverage we often have to consider "weighted averages" of n quantities11' ... ,In givenbyPIll+ P2!2 + ... +pnfn=/1,PI+ P2+ ... + pnwherethe"weight factors" Pi areanypositivequantities. If, for example,PI' P2, ... ,pnare actuallythe weights of particles located respectively atthepoints 11,/2' ... ,Inof the x-axis, then I ~ will represent the location ofthecenterof gravity. IfaUweightsPi areequal, thequantityIt isjust thearithmetic average definedabove.For a function f(:l') we can formanalogously theweightedaverage(13)IIIa I(x)p(.r) d:tI'}p(.f)dol'nover the interval[a, h] where p(.r), theweight jllnction, is any positive functionin the interval. The assumption that Pis positive guarantees that thedenonlinator does notvanish.The weighted average Ii also lies betweenthe largest value M and the smallestvalue m l ~ l the ./imction f inthe interval.For multiplyingtheinequalitym s./{x) sM,bythe positil'enumber p(x), wefindthatmp(.r) S f(:l:)p(.r) SMp(.r).IntegrationthenyieldsmIbp(.r)dov Sif(J:)P(X) dxSM(bp(.t) dr.a a JaDividing by thepositive quantity I"p(,,) dx, we indeedobtainthe resultmSfl sM.If heref(x) is continuous, we conclude fromthe intermediate valuetheorem(p. 44) that p, = f ( ~ ) , where ~ is a suitable value in the intervalaS ~ Sb. This leads tothefollowinggeneralizedmeanvaluetheorem ofintegral calculus:Sec. 2.4 The Integral as Functionof the Upper Limit (Indefinite Integral) 143If and p(:r) arecontinuousintheinterval [a, b]andmoreover p(;r) ispositive inthat interval, thenthere exists avalueinthe interval suchthat(14) d:l' =/mfp(I:) dx.The special case p(x) =1 leadsto our earliermeanvalue theorem.2.4 TheIntegral as a Functionofthe Upper Limit (IndefiniteIntegral)DefinitionandBasic FormulaThevalue of theintegral of afunction .(ex) dependsonthelimits ofintegration a and b: The integral is a function of the two limits a and b.In order to study this dependence on the limits more closely we imaginethe lower limit to be a fixed number, say denote the variable ofintegrationnolongerbyx butby u (seep. 126), anddenotetheupperlimit by xinstead of by b inorder toindicate thatwe shaH consider theupperlimit as thevariableandthat wewishtoinvestigatethe valueoftheintegral as a functionofthis upper limit. Accordingly, wewriteep(x) =f/(u) duoWe call the function 4>(x) an indefiniteintegral ofthe function I(x).Whenwespeak of an andnot of theindefinite integral, we suggest thatinstead of the lower limit (f..any other could be chosen,in which case weshouldordinarilyobtainadifferent valuefor theintegral. Geometri-cally, the indefinite integral 4>(x) is givenby the area (shownby shadingin Fig. 2.17) under the curveY =.l(u) and bounded by the u-axis,the ordinateu = and the variable ordinate u =x, the sign beingdeterminedbytherulesdiscussedearlier(p. 126).Anyparticulardefiniteintegral isfoundfromtheindefiniteintegral4>(x). Indeed, byourbasicrulesforintegrals,f/(U) du= ff(U)du +ff(U)du= - f/(U) du +f/(U) du= ep(b)- ep(a).]nparticular, we canexpressanyotherindefiniteintegralwithalowerlimit interms of 4>(x):ffeu)du =ep(x) - ep(oc').144 TheFundamental Ideasof the Integral andDifferential Calculus Ch.2yFigure 2.17 The indefinite integral as an area.As we see, any indefinite integra! differs from the special indefiniteintegral (x)only by a constant.Continuityofthe Indefinite IntegralIf thefunction I(x) is continuous inthe interval [a, b] and fY. is apoint of that interval, then the indefinite integral,p(x)= ff(U)durepresentsafunctionof xwhichisagaindefinedin thesameinterval.As easilyseen: The indefinite integral (x) of acontinuousfunctionf(x) is likewisecontinuous. For if xandyareanytwovaluesintheintervalwe haveby the meanvalue theoremthat(15) ,p(y) - ,p(x) = ff(U)du= f ( ~ ) ( y - x)where~ is some value in the interval with end pointsx and y. From thecontinuity ofI wehave thenlim(y)=lim[(x) +f ( ~ ) ( y - x)] = (x) +f(x) 0=(x),whichshows that eP is continuous. More specifically, inanyclosedinterval we haveleP(Y) - (x) 1 sMIy - xl, whereMis the maximumof IfIinthe interval, sothat r/>is evenLipschitz-continuous.Sec. 2.5 LogarithmDefinedbyanIntegral 145Formula (15) for>(y) - >(x) shows: that >(x) isanincreasingfunc-tion ofj; in case f is positive throughout the interval, namely, for y > x>(y) =>(x) +f ( ~ ) ( y - x) >>(x).Formingtheindefiniteintegral of afunctionis animportantway ofgeneratingnewtypesof functions. I nSection 2.5we shall applythismethodtointroducethelogarithmfunction. Thiswill alsogiveus afirst glimpse of thefact that general theorems of mathematical analysisleadtothemost remarkable specificformulas.As we shall see in Section 3.14a (p. 298), the definition of newfunctions by means of integrals of already defined functions is asatisfactoryprocedureif wewishto put definitions (for example, of thetrigonometricfunctions) onapurelyanalytical basisinstead of relyingonintuitive geometrical explanations.2.5 LogarithmDefined by anIntegrala. Definition of theLogarithmFunctionIIIIn Section 2.2 we had succeeded in expressing a XCIdx for any rationalex yE:. -1interms ofpowers ofaandb. For ex = -1we were onlyabletorepresent theintegral aslimit of asequenceIII 1 d J' lI/-l/- U= 1mn( v) a- I).a U n-+ocJIndependentlyofthe discussions ofSection2.2wenowintroducethefunction representedbytheindefiniteintegralJX1- dU,l1 Uor, geometrically, bythearea undera hyperbolaas indicatedinFig.2.18. We call it the logarithmofX, or more accurately the naturallogarithnl of x, andwrite( 16)f,X1logX = - duo1 USince y= llu is a continuous and positive function for all u > 0,the functionJog xisdefined forallx > 0, is moreover continuous, andalso is monotonicallyincreasing. ThechoiceofI asthelower limitin1 In this section we again freelyuse the fact that the integral of a continuous function(here thefunctionl/u) exists; the general proof is giveninthe Supplement.146 The Fundamental Ideas 0/ theIntegral andDifferential Calculus Ch.2y,\\\Y=1z\\\\\'\o x(18)oFigure 2.18 Log xrepresented byanarea.the indefinite integral for log x is a matter of convenience. It implies that(17) log 1 =0,and that log x is positive forx >1 and negative forxbetween zero and] (Fig. 2.19). Anydefinite integral of Iju between positive limits aand bcan be expressed in terms of logarithms by the formula (see p. 143)If) 1- du= log b - log a.a Uy X4 5-1-2Figure 2.19 The natural logarithm.Sec. 2.5LogarithmDefinedb)' an Integral 147Geometrically, this integral represents the area under the hyperbolay =lfxbetweentheordinatesx =aandx =b.h. TheAdditionTheorem forLogarithmsThefundamental propertywhichjustifies the traditional name forlogxis expressedbytheADDITIONTHEOREM. Forany positivexand y(19) log (xy) = log x + log y.PROOF. Wewritetheadditiontheoreminthe formlog Cry) - log y= log xorJ, XlI1 fX1- dv= - du,Y V 1 Uwherewehavedeliberatelychosendifferentlettersfor thevariablesofintegrationinthetwointegrals. Theequality of thetwointegralswillfollowfromthefactthattheapproximatingsumshavethesamevaluefor suitable choices of subdivisions and of intermediate points. Assumeat first x >1. Thenfx 1 n 1-du=lim L1 U whereUo = I, U1 , U2, , U11 = Xrepresent thepointsarisinginasub-division of theinterval[1, :r] and lies in the ith cell. Puttingl"i = YUi'1/i=we see that the points (1o, l11 , , I1 ncorrespond to a sub-division of the interval [y, xy] with intermediate points 'YJi =Obviously,sothatFor n tending to infinity we obtain the desired identity between integralsforthe casex >1.For x = 1 the addition theoremholds trivially, since log1 = O.To prove the theorem also for the case 0 < x I, andhencelog x + log y= log x +log (; xy)1= log x +log - +log (xy)x1= log - + log x + log (xy)x= log (; x) +log (xy)=log1 + log (xy) =log (xy).This completestheproof of theadditiontheorem.Aproof of the addition theoremcan also be based on formula (3)(p. 134), according towhichlog x= lim- 1).Thenlog (xy) =lim n( - 1)n-OO 00=[lim - 1)] (lim +lim n(v/y - 1)n- OO= log x+ logy,since lim'\Iy= 1 (see p. 64).n-oo1&-00Applyingtheadditiontheoremtothe special case y= Ijxleads to1log 1 = logx + log -xor(20)1log - = -log x.x(21)More generally thenlog = log y +log! = log y- log x.x xRepeated application ofthe addition theoremto a product of nfactorsyieldslog (XIX2 xn) = log Xl +logX2+... +logXn.Sec. 2.6Exponential FunctionandPowers 149Inparticular, wefind that foranypositiveinteger n(22)log (x") =n log x.Thisidentityalsoholdsforn = 0, since XO =I, andcanbeextendedtonegativeintegers n byobserving thatlog (xn)=log C ~ , , ) =-log (x-n)= -(-n) logx =n logx.For any rational ~ = In/n and any positive a we can formaa =am!n= x. Wehavethen1 1 n1logx = -logxTl= -log am = - log a =~ log a.n n nThus theidentity(23)holdsforanypositiverealaandanyrational ~ .2.6 Exponential Functionand Powersa. TheLogarithm of theNumber eTheconstant e obtainedonp. 79asthelimit of (I +l/n)rl playsadistinguished role for the function log x. Indeed, the number e ischaracterizedbytheequationllog e = I.For the proofwe observe that the continuity of the function logximplieslog e = log [ ! ~ ~ (1 +~ n =! ~ ~ log [(1+~ n= lim n log (1 + 1.).n.....oo nNowbythemeanvaluetheoremof integral calculus(1') Jl+l!rl1 1 1log 1 + - = - du= - - ,n 1 U ~ n1 This means geometrically thatthe areaboundedby the hyperbola y =IJx and thelinesy = 0,x =1, andx = e hasthevalueone(seeFig. 2.18).150 The Fundamental Ideasof the Integral andDiflerential Calculus Ch. 2where ~ issome numberbetween1 and1 +linwhichdependson thechoice of n. Obviously, lim~ = 1 so thatn----.OO(24) log e = lim ! =1.n-+oo ~b. TheInverseFunction of the Logarithm. TheExponential FunctionFromtherelationlog e = 1 itfollowsthatfor anyrational rxlog (eel) = rx log e = rx.This shows that everyrational number rx occurs as a value oflog xfor some positive x. Since log x is continuous, it assumes then any valueintermediatebetweentworational values; this means all real values.It follows that for x varying over all positive values the values ofy =log x range over all numbers y. Since log x is monotonicallyincreasing, thereexistsfor anyreal yexactlyonepositivex suchthatlog x =y. Thesolutionx oftheequationy =log x is givenbytheinversefunctionof thelogarithmwhichweshall denotebyx= E(y).We knowthenthat E(y) (Fig. 2.20) isdefinedandpositivefor all y,and againcontinuous andincreasing (seep. 45)y----o----o-------o-----...x-2 -1 oFigure 2.20 The exponential function.Sec. 2.6Exponential FunctionandPowers 151Since the equations y=log x and x = E(y) stand for the same relationbetweenx andy, we canwrite the equation rJ.=log (e), whichis validforrational rJ., alsointhe formWesee: for anyrational rJ. thevalueof E( rJ.) is the rJ.th powerof thenumber e. For rational CI. = minthe power e is defined directlyas.yJem. For irrational l/.. theexpressionea. isdefined most naturallybyrepresenting r:J. as the limit of asequenceof rational numbers Cl.nandputting ea.=lim(ea. n). Since ea. n=E(iJ.,J and sincethefunction E(y)7/-"00depends continuouslyon y, we can be surethat the limit ofthe enexists and that it has the value E( CI.) independently of the specialsequence used to approxirnate l/... This proves that the equationE(l/..) = ea. holds for irrational CI. as well. For all real CI. wecannowwrite ea. instead of E(l/..). We call eJ' the exponential function. Thisfunctionisdefinedandcontinuousfor all x, isincreasing, andpositiveeverywhere.Since the equations y =log .r andx =eYaretwoways of expressingthesamerelationbetweenthenumbersxandy, wesee that log x, the"natural logarithm" ofx (as defined here byan integral) stands forthelogarithmtothebasee, asthat termwould be usedinelementarymathematics; that is, log xistheexponent of that power of e whichisequal toxor(25)Wecanwrite) log x = loge x.Similarly, x =eYisthat numberwhoselogarithmisy, or(26) log eY= y.Fromthe point of viewof calculus it is really easier to introducenatural logarithmsfirst asintegralsof thesimplefunctiony= llx, aswe didhere, andtodefine powers ofe bytakingthe inverseofthelogarithm function. In this way the continuity and monotonicity of thefunctionslog xandeXarisejust as consequencesof general theoremsandrequirenospecialarguments.1 The reader may feel thatthe name "natural logarithm" should havebeenreservedrather for logarithms tothe base 10. However, historicallythefirst tableoflog-arithms published by Napier in 1614essentially gave logarithms to the base e.Logarithmstothebase10 were introducedonlysubsequentlybyBriggsbecause oftheir obvious computational advantages.152 The Fundamental Ideasof the Integral andDifferential Calculus Ch. 2c. TheExponential Function as Limit of PowersOriginally weobtainedthenumber e as the limite =lim (1 + !)n.n..... oo nAmore general formularepresentseX foranyxasa limit(27) egJ =lim (1 + n..... oo nFor the proof it is sufficient toshow that the sequenceSn=log (1 + hasthe limit x. For then the sequence of valuese'n= (1 + must tend to eX sincetheexponential functionis continuous. NowSn = n log(1 += nf+x,n Bythemeanvalue theorem of integral calculuswehave5n =n .l.[(1+ :) - IJ= ;-,n ;nwhere n is some value between one and 1 + x/no Since obviouslyntendstoone for n tending to 00, wehave indeedlim sn= X.11-00d. DefinUion of Arbitrary Powers of PositiveNumbersArbitrarypowersof anypositivenumberscannowbeexpressedinterms of the exponential andlogarithmic functions.1We foundforrational r:J..andanypositivexthat the relationlog (x) =alogxholds. Wewrite this equation inthe formx= erl.lOga:.1 Thisobviatesthemoreclumsy definitionand justificationof theseprocesses by passage to thelimit from rational exponents indicated on p.86.Sec. 2.6Exponential FunctionandPowers 153For irrational rx weagainrepresent rx aslimit of asequence of rationalDumbers rxnanddefineXCX =limXCXn=lim ecxnlog x.1/. .... 00 n-+ooThe continuity of the exponential function implies again that thelimit exists and that it hasthevalueea.log x, sinceHencetheequation(28)holdsquitegenerallyforany rx andanypositivex. Puttinglog x ={3or, what isthesame, x =efJwe infer(29) (eP)CX =ecxP,andmore generallythenforanypositivex(xa.)p= (eCX log X)p =ecxPIOr;l; x =xcxP.Another rule for workingwith powers whichis easilyestablishedincompletegenerality, isthemultiplicationlawlog(XCXxP) = log (xa.+p).Nowby therules(19), (26), and(28) alreadyestablishedit follows thatlog (xcrxP) = logxa. + logxfJ= log (ea.lOgx) + log (ePIOg x)= rx logx + f3 logx =(rx +fJ) logx=log (e(cr+P) log X) =log (xa+fJ).wherexisapositivenumber andrxandf3arearbitrary. It issufficienttoprove the corresponding formulaobtainedbytaking thelogarithmsof bothsides:e. Logarithms to Any Basey =l o ~ x .It is easytoexpress logarithms toa baseother thaneinterms ofnaturallogarithms. If for apositivenumberatheequationx =aUissatisfied, wewrite(30)Nowa'Y= e1J laga, so that x= e1J log aor y log a = log x. It follows thatlogxloga x = --,log a154 The Fundamental Ideas of the Integral andDifferential Calculus Ch.2wherelog xisthenatural logarithmtothebasee. Inparticular, thecommonlogarithms tothebase10 aregivenbylog x]OgIOX=--.log 10Since logarithms to any base a are proportional to natural log-arithms, theysatisfythesameadditiontheorem:2.7 TheIntegral ofan ArbitraryPower ofxInSection 2.2weobtainedtheformulailJa ba+l- aai 1U du= ,a ~ +1for anyrational ~ ~ -1. (Thecase ~ =-1wasseentoleadtothelogarithm.) To evaluate theintegral when (J. isan irrational number, itis sufficient to discusstheindefiniteintegralq,(x)= fU dufromwhichall definite integrals with positive limits aandbcanbeobtained. Assumex >1 (thecasex 0or f'(x) < 0 throughout the interval,then theinverse function x =4>(y) alsopossesses aderh1ative at everyinteriorpoint of its intervalof definition: the derivativesof y = f(x)and of its inverse x =4>(y) satisfy the relationf'(x). >'(y) =1 atcorresponding values x, y.Thisrelation canalsobe put in theform(5) dy =.ldx dxdyThis last formula again illustrates the suitability of Leibnitz's notation:thesymbolicquotient dyldxcanbetreatedinformulas as ifit wereanactualfraction.PROOF. The proof of thetheorem is simple. Writing thederivativeasthelimit of adifference quotient, wehavey' =j'(x)=limt::.y = limYI - Y,&x-o t::.x Xl-ta: Xl - xwherexand Y = f(x), andXlandYI = f(xl ), respectivelydenotepairsof corresponding values. By hypothesis the first of these limiting valuesis not equal to zero. Because of the continuity of y = f(x) and x =cP(y),therelationsYI -+ Y andXl -+X are equivalent. Therefore the limitingvaluelimXl - x =lim Xl - xXl .... X YI - Y 111 .... 11 YI - Yexists andis equalto1If'(x). On the other hand, the limiting value ontheright-handsideis bydefinitionthederivative 4>'(y) of theinversefunction(y) makes anangle(I.with the positive x-axis, and an anglef3 with the positive y-axis;fromthegeometrical interpretationof thederivativeof afunctionastheslopeof thetangentf'(x) =tan(I., 4>'(y) =tan p.Since the sum of the angles (I.and pisTr12, tan(J..tan p=1, and thisrelationshipis exactly equivalent toour differentiation formula.208 The Techniquesof CalculusyyaChi 3-------"---.....!---o-+----.....I-----....... rFigure 3.1 Differentiation of the inverse function.Critical PointsWe have hitherto expressly assumed that either f'(x) > 0 orf'(x) < 0, that is, that f'(x) is never zero. What, then, happens iff'(x) = O? If f'(x) = 0everywhereinaninterval, then! is constantthere, andconsequentlyhas noinversebecausethesamevalueofycorresponds to all values ofx in the interval. If !'(x)= 0 onlyatisolated"critical"points(andif!'(x)is assumedcontinuous), thenwehavetwocases, accordingtowhetheronpassing throughthesepointsf'(x)changes sign, or not. Inthe first case thispoint separates a pointwherethefunctionismonotonicincreasingfromanother whereit ismonotonic decreasing. In the neighborhood of such a point therecan be no single-valued inverse function. In the second case thevanishing of the derivative does not contradict the monotonic characterof thefunctiony=f(x), sothat a single-valuedinverseexists. How-ever, the inverse function is no longer differentiable at the corre-spondingpoint; infact, itsderivativeis infinitethere. Thefunctionsy = x2andy= x3at the point x= 0 offer examples of thetwo types.Figure 3.2and Fig. 3.3illustrate the behavior ofthe twofunctionsuponpassingthroughtheoriginandat thesametimeshowthat thefunctiony = x3has a single-valued inverse, whereas the other functiony= x2does not.Sec. 3.2The Derivativeof the Inverse Function 209o210 The Techniques of Calculus Ch.3b. TheInverse of the nthPower: the nthRootThe simplest example is the inverse of the function y= xnforpositive integers n; at first weassumepositive values of x, hencealsoy > o. Under theseconditions y' is always positive, sothat for allpositive values of y we can formthe unique inverse functionx = vry = yl/n.Thederivativeof thisinversefunctionis immediatelyobtainedbytheabove general rule as follows:d(yl/n)= dx=_1_=_1_ =1_1_ = ! y(l/n)-l.dy dy dyjdx nxn-1ny(n-U/n nIf we nowchange the notation and denote again the independentvariableby x, wemay finallywrited'\Y;=.!!- (x1/n) = ! x(l/n)-l,dx dx nwhich agrees withthe result obtained onp. 164.For n >1, the point x= 0 requires special consideration. If xapproaches zero through positive values, d(xl/n)jdx will obviouslyincrease beyond all bounds; this corresponds to the fact that forn >1 the derivative of the nthpower I(x)= xnvanishes at the origin.Geometrically, this meansthat the curves y= xl/nforn >1 touch they-axis at theorigin (cf. Fig. 1.35, p. 48).Itshouldbenotedthatfor oddvaluesof ntheassumptionx > 0can be omitted and the function y= xnis monotonic and has aninverse over the entiredomain of real numbers. The formulade fly) = (l!n)y(l/nl-ldystill holds for negative values ofy, but for x =0, n >1, we haved(xn)jdx= 0, whichcorrespondstoaninfinitederivativedxjdyof theinverse functionat the point y= 0.c. The InverseTrigonometric Functions-MultivaluednessToformtheinversesof thetrigonometricfunctions weonceagainconsiderthegraphs1of sin x, cos x, tan x, andcot x. Weseeat oncefromFigs. 1.37, p.50 and1.38, p. 51, that for each of these functions it1 The graphical representation wiJI help the reader to overcome the slight difficultiesinherent inthe discussionof tbe "multivaluedness" of the inverse functions.Sec.3.2 The Derivativeof the Inverse Function 211isnecessarytoselectadefiniteinterval if wearetospeak of auniqueinverse; forthelinesy = cparallel tothex-axiscut thecurvesinaninfinite number of points, if atall.The Inverse Sine andCosineFor the function y =sin x, for example (Fig. 3.4), the derivativey' =cos xispositive in the interval -7T/2 (lp(X)] isa compound function of its derivativeisgivenbytheruledy =y' = g'(u)4>'(v)lp'(x) = dy. du. dv ;dx du dv dxsimilar relations are true for functions that are compounded ofanarbitrarynumber of functions.Higher Derivatives of aCompositeFunction. y =g[ep(x)] can be foundeasilybyrepeated applicationof the chain rule andthe preceding rules:dydc/>y' = -- =g'ep'ddx 'y" =g"ep'2 +g'ep",y'" =g",'3 +3g"c/>'c/>" +g,c/>/I1.Analogous formulasfory"ll etc., can be derived successively.220 The Techniquesof CalculusCh. 3Finally, let us examine the composition oftwofunctions inverse to eachother. The function g(y)is the inverse of y =4>(x)ifI(x) =g[4>(x)] =x. It followsthatf'(x) = g ' ( y ) ~ ' ( x ) = 1whichis exactly theresult of Section3.2, p. 207.Examples. Asasimplebut important example of anapplication ofthe chainrulewe differentiatexa.(x > 0) foranarbitrary real power oc.In Chapter 2, p. 152, wedefinedg'(y)=eV 1p'(U)= (1",wealsoproved for ~ ( x ) = log x, 1p(u)= (1"U, g(y)=eVthat>'(x)= ! ,xNowxa. is the compound functiong{1p[>(x)]}. Applying thechainruleweobtain the general formulad- (xa.) = g'(y)1p'(u)ep'(x)dx=e'IJ(1".!xrJ.ea.logxxhence~ (xa.) = (1"xa.-l,dxa result we couldprove onlywithsome difficultyhadwe attemptedtoproceed directly from the definition of xa for irrational (1" as the limitof powerswithrational exponents.An immediate consequence of this differentiation is, again, theintegral formula(1" ~ -1).Asasecond example, we consideror y = #,Sec. 3.3 DifJerentiationof Composite Functions 221wherecP= 1 - x2and -1 < x 0, thecurveintheneighborhood of the point lies above the tangent while when f"(x) < 0,1 Wemakeusehereoftheintuitivelyobviousobservation: acontinuous functiong(x) whichis positiveat apoint Xo alsois positivefor all points of asufficientlysmall neighborhoodofXo (asfar astheybelongtothedomainof g). The formalproof issimple. Fromthecontinuityof gat Xo weknowthatforeverypositivethe inequality Ig(x) - g(xo)1 < holds forallxin a sufficiently small neighborhoodIx- xol< b of the point xo. Since g(xo) > 0, we are freeto choose for the value19(xo), sothatIg(x) - g(xo)1 < ig(xo) insomeneighborhood. Sincetheng(xo) -g(x) ~ Ig(x) - g(xo)! < 19(xo), ~ t followsthat g(x) > ig(xo) > O.Sec. 3.6y-o-+--.;.-----ox--...... xMaximaandMinima 237y- o : : : - t - - - - - - - < > x - - - ~ xw WFigure 3.14 (a) ("(x) >O. (b) ["(x) < O.it lies below the tangent (see Figs. 3.14a and 3.14b) (cf. Problem 4, p. 200and Section5.6).Point oj'InflectionSpecial consideration is required only in points wherej"(x) =O.Onpassingthroughsuchapointthesecondderivative ["(x) will gen-erallychangeitssign. Suchapointwill thenbeapoint of transitionbetween the two cases just indicated; that is, on one side the tangent isabovethecurveandonthe other side below it, whereas at this point itcrossesthecurve(seeFig. 3.15). Suchapoint iscalleda point of in-flection ofthecurve, and thecorrespondingtangent is calledanin-flectionaltangent.yoFigure 3.15 Point of inflection.238 The Techniquesof CalculusCh. 3Thesimplest exampleis givenbythe functiony= x3, thecubicalparabola, for whichthex-axis itselfis aninflectional tangent at theinflection point x= 0 (seeFig. 3.3, p. 209). Another example isgivenbythe functionj{x) = sin x, forwhichf'(x) = d(sin x)jdx =cos x and f"(x) =d2(sin x)jdx2= - sin x.Consequently,j'(O)= 1 anqj"(O)= 0; since the sign ofj"(x) changesat x=0, the sinecurve has at the originaninflectional tangent in-clined at an angle of 45degreestothe x-axis.It must,however, be noted that points can exist where f"(x) =0 andthesigno(f"(x)doesnotchangewithincreasingx, whilethetangentdoes not cut the curve but remains entirelyon one side ofit. Forexample, thecurvey= x4liesentirelyabovethex-axis, althoughthesecondderivative f"(x) =12x2vanishes forx = o.b. Maxima and Minima-RelativeExtrema. Stationary PointsAfunction f(x) hasamaximumatapoint ~ if thevalue off atthepoint ~ isnotexceededbythevalueof f at anyotherpoint xof thedomain off;that is,f(;) z.f(x) for all x wherefis defined.1Similarly,fhasa minimum at ~ iff ( ~ ) :::;;; f(x)forallxin thedomain. Thewordextrema isusedto cover bothmaxima andminima.The functionf(x) = J1 - x2, for example, which is defined for-1~ x ~ 1, hasminima atx =1andamaximumatx =O. It iseasytogiveexamplesof continuous functionswhichhavenomaximaor no minima. Thus the function/(x) =1/(1 + x2) (Fig.3.8, p. 216) inthe domain- 00 < x < +00 has nominimum; thefunction I(x)=l/x definedfor0 < x f(x)for all x inthe domain off thatare different from~ .Sec. 3.6MaximaandMinima 239hasits greatest (least) value at ~ when compared notwithall possiblevalues of/(x) but just with the values of/(x) for x in some neighborhoodof~ . By a neighborhood of the point ~ we mean here any open interval~ < x < fJ whichcontains thepoint ~ but maybearbitrarilysmall.A relative extremum point ~ ofI is then a point which is an extremumpoint when f isrestrictedtoall those pointsof itsdomainlyingsuf-ficientlycloseto ~ . 1 Obviously, theextremaof thefunctionarein-cludedamongtherelativeextrema. ToavoidconfusionweshalluseyxFigure 3.16 Graph of function defined on the interval [a, b] with relative minima atx = a,X hx4, X 6, relative maxima atX hXs, XII'b, absolute maximum at b, and absoluteminimum at X4theterms absolutemaxima(minima) for themaximaandminimaoff in its entire domain (seeFig. 3.16).Geometricallyspeaking, relative maxima andminima, if not locatedin the end points of the interval of definition,are respectively the wavecrestsandtroughs of thecurve. A glanceat Fig. 3.16 showsthat thevalue of a relative maximum at one pointX6 may very well be less thanthe value of a relative minimum at another point x2 The diagram alsosuggests thefact that relative maximaandminimaofacontinuousfunctionalternate: Betweentwosuccessiverelativemaximathereisalways located a relative minimum.Let f(x) beadifferentiablefunctiondefinedintheclosedintervala ~ x ~ b. We see at once that at a relative extremum point which is1 The formal definition ofarelativemaximum point ewould state that there existsanopeninterval containing esuchthat f() ~ f(x) for all x ofthat interval forwhich fis defined.240 The Techniquesof CalculusCh.3located inthe interior of the intervalthetangenttothe curvemustbehorizontal. (Theformal proof isgivenbelow.) Hencetheconditionf'(,) = 0is necessaryfor a relativeextremumat the point , witha < , < b.If, however, I(,) is a relative extremum and, coincides with one of theend points of the interval of definition, the equationf'(,)=0 need nothold. Wecanonlysaythat ifthe left-handendpoint is a relativemaximum(minimum) point, the slope f'(a) of the curve cannot bepositive(negative), while ifthe right-hand end point bis a relativemaximum (minimum) then.r'(b) cannot benegative (positive).Thepoints at which thetangent tothe curvey=f(x)ishorizontal,correspondingtotherootsof theequation f'(,) = 0, arecalledthecritical points or stationary points of f All relative extrema of adifferentiablefunction f whichareinterior pointsof thedomainof fare stationary points. Hence: an absolute maximum or minimum of thefunction coincides either with a criticalpoint of thefunction or withanend point of itsdomain. Inorder tolocatetheabsolute maxima(minima) of the function we have only to compare the values of/in thecritical points and in the two end points and to see which of these valuesaregreatest (least). If / fails at afinite number of points tohaveaderivative, we have onlyto add those points to the list ofpossiblelocationsof anextremumandtocheckalsothevalues of f at thosepoints. Thusthemainlabor indetermining the extrema of a functionis reduced to that of findingthe zeros of the derivative of the function,which usuallyare finitein number.Totakeasimpleexample, let usdeterminethelargest andsmallestvalues of the function/(x) = l-ox6- lox2in the interval -2 Sx:s;; 2.Here the critical points, the roots of the equationf'(x) =6(x5- x)!IO=0arelocatedat x=0, +1, -1. Computingthe values of f at thosepoints andalso at the endpoints of the interval, we findx -2 -1 o 2lex) 5.2 -0.2 0 -0.2 5.2Itisclearthat thepointsx= 1 represent relativeminima, whereasrelativemaxima occur at x= 0 andx= 2. Themaximumvalue ofthefunction, assumedinthe end points ofthe interval, is 5.2; theminimum value, assumed in the points x= 1, is -0.2 (see Fig.3.17).Without appealing to intuition we can easily prove by purely analyticmethods= 0 whenever, is arelative extremum point intheinterior of the domain off provided f isdifferentiable at (CompareSec. 3.6MaximaandMinima 241y654-1Figure 3.17 y= (x8- 3x2)jlO.theexactlyanalogousconsiderationsfor Rolle'stheorem, p. 175.) Ifthefunction I(x) hasarelativemaximumat thepoint ~ , thenfor allsufficiently small values of h different from zero the expressionf ( ~ + h) - f ( ~ ) must be negative or zero. Therefore[ f ( ~ + h) - f(;)] < 0h -forh > 0, whereas[ f ( ~ + h)- f(x)]h 20for h < O. Thusif htends tozerothroughpositivevalues, thelimitcannot bepositive, whereas if h tendstozero throughnegative values,the limit cannot be negative. However, since we have assumed that thederivativeat ~ exists, thesetwolimitsmust beequal tooneanother,and, infact, tothevaluef ' ( ~ ) , whichthereforecanonlybezero; wemust havef'(;) =O. A similar proof holds for a relative minimum. Theproofalsoshows that ifthe left-handendpoint ; =ais a relativemaximum(minimum)point, then atleastf'(a) ~ 0[f'(a) ~ 0]; if theright-handendpoint b isarelativemaximum(minimum)point, then.['(b) ~ 0 f/'(b) =:;: 0].242 The Techniquesof CalculusCh.3Thecondition = 0characterizingthecritical points is bynomeans sufficient for the occurrence of a relative extremum. There maybe points at which the derivative vanishes, that is, at which the tangentishorizontal, althoughthecurvehasneitherarelativemaximumnorminimumthere. This occurs ifat the given point the curve has ahorizontal inflectional tangent cutting it, as in the example of thefunctiony = x3atthepointx = o.Thefollowingtest givestheconditionsunderwhichacriticalpointis a point of relative maximum or minimum. It applies to a continuousfunction f, havingacontinuous derivative f' whichvanishes at mostat afinite number of points or, more generally to differentiable functionsIforwhich I'changes sign at most at afinitenumber of points:The function f(x)has a relative extremum at an interior pointof itsdomain and only if, the derivativef'(x) changes sign as x passes throughthis point, in particular, the functionhasarelativeminimum if near the derivative is negative to the left oj'and positive to the right, whereasinthe contrary case it has a maximum.Weprovethisrigorouslybyusingthemeanvaluetheorem. First,we observe that to the left and right ofthere exist intervals < X 0for x >andy' < 0for x < 0(cf. Fig. 2.24, p. 167).Thefunction y={1;2likewise has a minimumat the point x= 0,eventhoughits derivative isinfinitethere (cf. Fig. 2.27, p. 169).Sec.3.6MaximaandMinima 243The simplest method for deciding whether a critical pointis arelatiye maximumor minimuminvolves thesecond derh'ativeat thatpoint. It is intuitively clear that if = 0, then I has a relativemaximum atif < 0, andarelativeminimum if >O. Forinthefirst casethecurveintheneighborhoodof this pointliescom-pletelybelow the tangent, andin thesecondcase completely abovethetangent. This result follows analytically from the preceding test,providedthat f(x) and f'(x) arecontinuousandthatexists. For = 0 and,> 0, wehave =lim+h) -=lim+h) >O.11.-+0 h h-+O hIt follows that + h)jh > 0 for all hr!:. 0 which are sufficientlysmall inabsolute value; hence f' +h) and h havethesame sign in aneighborhood For xnearthederivative j'(x)mustbenegativefor x to the left of andpositive for x to the right of thisimpliesthat thereisarelativeminimumatThesituationis particularlysimpleincase f"(x) is ofoneandthesamesign throughout theinterval [0, b]inwhich f isdefined:A pointat which f' vanishesisamaximum point ofI if j"(x) < 0throughout the interval (or ifits curve is concave), and 0 minimumpoint oj'f if throughout the interval f"(x) > 0(that is, if the curve isconvex).Indeed, if j"(x) < 0 the functionI'(x) is monotonic decreasing,hence hasas its only zero. Moreover,f'> 0 for a x 1,for Ixl< ~ .2Someoftheearlier examples involvingthe symbol 0 cannowberefinedtoindicateabetterestimateof theerror withthehelpof thesymbolO. Thuswehave forafunctionfforwhichf"isdefinedandcontinuousf(x + h)- f(x) =hf'(x) +O(h2)Other examples arefor h -+O.1 _2- + o(!-)J1 + 4x2- 2x x2'cosX= 1 + O(x2) forall x.The same notations can be used for sequences an, letting the index ntend to infinity. We shall meet some interestingexamples ofsuch"asymptotic" formulaswith an error term of higher order in the sequel(cf. Stirling'sformulafor 11! onp. 504). A famousasymptotic law,21Notice that f =O(g) does not mean that fighas the limit one or that the quotientnecessarilyhas anylimit at all.I Theproof cannot begiveninthisbook. SeeA. E.Ingham, The DistributionofPrimes, Cambridge University Press,1932.Sec. A.ISomeSpecial Functions 255alreadymentionedinChapter