IntrotoCalculus(Index)

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lassicsin MathematicsRichard Courant Fritz John Introduction to Calculus and nalysis

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Springer V erlag Berlin Heidelberg GmbH


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Richard Courant Fritz John

ntroductionto Calculus nd nalysisVolumeReprint of the 989 Edition

pringer


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Richard Courant Fritz John

Introduction toCalculus and nalysisolume I

With 204 Illustrations

Springer Science Business Media, LLC


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IS N 978 3 540 65058 4 IS N 978 3 642 58604 0 eBook)OI 10.1007/978 3 642 58604 0


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reface

During the latter part of the seventeenth century the new mathematical analysis emerged as the dominating force in mathematics.t is characterized by the amazingly successful operation with infiniteprocesses or limits. Two of these processes, differentiation and integration, became the core of the systematic Differential and IntegralCalculus, often simply called Calculus, basic for all of analysis.The importance of the new discoveries and methods was immediatelyfelt and caused profound intellectual excitement. Yet, to gain masteryof the powerful art appeared t first a formidable task, for the available publications were scanty, unsystematic, and often lacking inclarity. Thus, it was fortunate indeed for mathematics and sciencein general that leaders in the new movement soon recognized thevital need for writing textbooks aimed t making the subject accessible to a public much larger than the very small intellectual elite ofthe early days. One of the greatest mathematicians of modern times,Leonard Euler, established in introductory books a firm tradition andthese books of the eighteenth century have remained sources of inspiration until today, even though much progress has been made in theclarification and simplification of the material.

After Euler, one author after the other adhered to the separation ofdifferential calculus from integral calculus, thereby obscuring a keypoint, the reciprocity between differentiation and integration. Only in1927 when the first edition of R. Courant's German Vorlesungen iiberDifferential und Integralrechnung appeared in the Springer-Verlagwas this separation eliminated and the calculus presented as a unifiedsubject.From that 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 ~prepared and published by Blackie and Sons in Glasgow since 1934 and

v


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vi refacedistributed in the United States in numerous reprintings by Interscience-Wiley.During the years it became apparent that the need of college and university instruction in the United States made a rewriting of this workdesirable. Yet it seemed unwise to tamper with the original versionswhich have remained and still are viable.Instead of trying to remodel the existing work it seemed preferable tosupplement it by an essentially new book in many ways related to theEuropean originals but more specifically directed at the needs of thepresent and future students in the United States. Such a plan becamefeasible when Fritz John who had already greatly helped in the preparation of the first English edition agreed to write the new book togetherwith R. Courant.While it differs markedly in form and content from the original it isanimated by the same intention: To lead the student directly to theheart of the subject and to prepare him for active application of hisknowledge. t avoids the dogmatic style which conceals the motivationand the roots of the calculus in intuitive reality. To exhibit the interaction between mathematical analysis and its various applications and toemphasize the role of intuition remains an important aim of this newbook. Somewhat strengthened precision does not as w hope interfere with this aim.Mathematics presented as a closed linearly ordered system of truthswithout reference to origin and purpose has its charm and satisfies aphilosophical need. But the attitude of introverted science is unsuitablefor students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation andatrophy of mathematics. t seems extremely important that studentsand instructors should be protected from smug purism.The book is addressed to students on various levels to mathematicians scientists engineers. t does not pretend to make the subjecteasy by glossing over difficulties but rather tries to help the genuinelyinterested reader by throwing light on the interconnections and purposesof the whole.Instead of obstructing the access to the wealth of facts by lengthydiscussions of a fundamental nature w have sometimes postponed suchdiscussions to appendices in the various chapters.Numerous examples and problems are given at the end of variouschapters. Some are challenging some are even difficult; most of themsupplement the material in the text. n an additional pamphlet more


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Preface viiproblems and exercises of a routine character will be collected andmoreover answers or hints for the solutions will be given.Many colleagues and friends have been helpful. Albert A Blanknot only greatly contributed incisive and constructive criticism but healso played a major role in ordering augmenting and sifting of theproblems and exercises and moreover he assumed the main responsi-bility for the pamphlet. Alan Solomon helped most unselfishly andeffectively in all phases of the preparation of the book. Thanks is alsodue to Charlotte John Anneli Lax R. Richtmyer and other friendsincluding James and Virginia McShane.The first volume is concerned primarily with functions of a singlevariable whereas the second volume will discuss the more ramifiedtheories of calculus for functions of several variables.A final remark should be addressed to the student reader. t mightprove frustrating to attempt mastery of the subject by studying such abook page by page following an even path. Only by selecting shortcutsfirst and returning time and again to the same questions and difficultiescan one gradually attain a better understanding from a more elevatedpoint.

n attempt was made to assist users of the book by marking with anasterisk some passages which might impede the reader t his first at-tempt. Also some of the more difficult problems are marked by anasterisk.

We hope that the work in the present new form will be useful to theyoung generation of scientists. We are aware of many imperfectionsand we sincerely invite critical comment which might be helpful for laterimprovements.

June 965Richard ourantritz John


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Contents

Chapter Introduction1.1 The Continuum of Numbersa. The System o Natural Numbers and Its

Extension. Counting and Measuring 1b Real Numbers and Nested Intervals 7c. Decimal Fractions. Bases Other ThanTen 9 d Definition o Neighborhood 12e. Inequalities 12

1

1.2 The Concept of Function 17a. Mapping-Graph 18 b Definition o theConcept o Functions o a ContinuousVariable. Domain and Range o a Function 21c Graphical Representation. MonotonicFunctions 24 d. Continuity 31 e TheIntermediate Value Theorem. InverseFunctions 44

1.3 The Elementary Functions 47a. Rational Functions 47 b AlgebraicFunctions 49 c. Trigonometric Functions 49d The Exponential Function and theLogarithm 51 e Compound FunctionsSymbolic Products Inverse Functions 52

1.4 Sequences 55

1.5 Mathematical Induction 57ix


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x Contents1.6 The Limit of a Sequence 60

1 1 1a. a = ~ 6 1 b. a2m = a2m-l = 2m 62c. a. = ~ l 63 d. a. = 64ne. an = Of ',65f. Geometrical Illustration o the Limits oOf ' and 65 g. The Geometric Series 67h an = 69 i. n = Vn l v;; 9

1.7 Further Discussion of the Concept of Limit 70a. Definition o Convergence and Divergence 70b. Rational Operations with Limits 71c. Intrinsic Convergence Tests. MonotoneSequences 73 d. Infinite Series and theSummation Symbol 75 e. The Number e 77f. The Number 7r as a Limit 80

1.8 The Concept of Limit for Functions of a Con-tinuous Variable 82a. Some Remarks about the ElementaryFunctions 86

Supplements 87S.1 Limits and the Number Concept 89a. The Rational Numbers 89 b. Real

Numbers Determined by Nested Sequences oRational Intervals 90 c. Order Limits andArithmetic Operations for Real Numbers 92d. Completeness o the Number Continuum.Compactness o Closed Intervals. ConvergenceCriteria 94 e. Least Upper Bound andGreatest Lower Bound 97 f. Denumerabilityo the Rational Numbers 98

S.2 Theorems on Continuous FunctionsS.3 Polar CoordinatesS.4 Remarks on Complex NumbersPROBLEMS

99101

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Contents xi

Chapter The Fundamental Ideas of the Integraland Differential Calculus 1192 1 The Integral 120a. Introduction, 120 b. The Integral as an

Area, 121 c. Analytic Definition of theIntegral. Notations, 122

2.2 Elementary Examples of Integration 128a. Integration of Linear Function, 128b Integration of x2 130 c. Integration ofxa for Integers a r -1 131 d. Integration ofx for Rational a Other Than -1 134e. Integration of sin x and cos x, 135

2.3 Fundamental Rules of Integration 136a. Additivity, 136 b. Integral of a Sum of aProduct with a Constant, 137 c. EstimatingIntegrals,138, d. The Mean Value Theoremfor Integrals, 139

2.4 The Integral as a Function of the Upper LimitIndefinite Integral) 1432.5 Logarithm Defined by an Integral 145a. Definition of the Logarithm Function, 145

b The Addition Theorem for Logarithms, 1472.6 Exponential Function and Powers 149a. The Logarithm of the Number e 149

b. The Inverse Function of the Logarithm.The Exponential Function, 150c. The Exponential Function as Limit ofPowers, 152 d Definition of ArbitraryPowers of Positive Numbers, 152e. Logarithms to Any Base, 153

2.7 The Integral of an Arbitrary Power of x 1542.8 The Derivative 155a. The Derivative and the Tangent, 156

b The Derivative as a Velocity, 162


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xii 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, 183

2.9 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, 191

Supplement The Existence of the Definite Integralof a Continuous Function 192

PROBLEMS 196

Chapter 3 The Techniques o Calculus 201Part Differentiation and Integration o theElementary Functions 2 13 1 The Simplest Rules for Differentiation andTheir Applications 201

a. Rules for Differentiation, 201b Differentiation of the Rational Functions, 204c. Differentiation of the TrigonometricFunctions, 205

3.2 The Derivative of the Inverse Function 206a. General Formula, 206 b The Inverse ofthe nth Power; the nth Root, 210 c TheInverse Trigonometric Functions


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ontents xiiiMultivaluedness, 210 d The CorrespondingIntegral Formulas, 215 e Derivative andIntegral o the Exponential Function, 216

3.3 Differentiation of Composite Functions 217a. Definitions, 217 b The Chain Rule, 218c. The Generalized Mean Value Theorem o theDifferential Calculus, 222

3.4 Some Applications of the ExponentialFunction 223a. Definition o the Exponential Function byMeans o a Differential Equation, 223b Interest Compounded Continuously.Radioactive Disintegration, 224 c. Coolingor Heating o a Body by a SurroundingMedium, 225 d Variation o theAtmospheric Pressure with the Height abovethe Surface o the Earth, 226 e. Progress o aChemical Reaction, 227 f Switching anElectric Circuit on or off, 228

3.5 The Hyperbolic Functions 228a. Analytical Definition, 228 b AdditionTheorems and Formulas for Differentiation 231c. The Inverse Hyperbolic Functions, 232d Further Analogies, 234

3.6 Maxima and Minimaa. Convexity and Concavity o Curves, 236b Maxima and Minima-Relative Extrema.Stationary Points, 238

236

3.7 The Order of Magnitude of Functions 248a. The Concept o Order o Magnitude. TheSimplest Cases, 248 b The Order oMagnitude o the Exponential Function and othe Logarithm, 249 c General Remarks, 251d The Order o Magnitude o a Function in theNeighborhood o an Arbitrary Point, 252e The Order o Magnitude or Smallness) o aFunction Tending to Zero, 252 f The 0and 0 Notation for Orders o Magnitude, 253


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xiv ContentsPPENDIX 255

A l Some Special Functions 255a. The Function y = ellx2 255 b. TheFunctiony = ellx 256 c. The Functiony = tanh l/x 257 d. The Functiony = x tanh l/x 258 e. The Functiony = x sin l/x y O) = 0, 259

A.2 Remarks on the Differentiability of Functions 259Part B Techniques of Integration3.8 Table of Elementary Integrals3.9 The Method of Substitutiona. The Substitution Formula. Integral of a

Composite Function, 263 b. A SecondDerivation of the Substitution Formula, 268c. Examples. Integration Formulas, 270

261263263

3.10 Further Examples of the Substitution Method 2713.11 Integration by Parts 274a. General Formula, 274 b. Further Examples

of Integration by Parts, 276 c. IntegralFormula for b) (a), 278 d. RecursiveFormulas,278 *e. Wallis s Infinite Productfor '11 ,280

3.12 Integration 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, 288

3.13 Integration of Some Other Gasses ofFunctions 290a. Preliminary Remarks on the RationalRepresentation of the Circle and theHyperbola, 290 b. Integration ofR(cos x, sin x), 293 c. Integration of


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Contents xvR cosh x, sinh x), 294 d Integration ofR x, ~ , 294 e. Integration ofR x, ~ , 295 f. Integration ofR x, -Vx2+1), 295 g. Integration ofR x, Vax2 2bx c ,295 h FurtherExamples of Reduction to Integrals of RationalFunctions, 296 i Remarks on the Examples,297

Part C Further Steps in the Theory of IntegralCalculus 2983.14 Integrals of Elementary Functions 298

a. Definition of Functions by Integrals. EllipticIntegrals and Functions, 298 b OnDifferentiation and Integration, 300

3.15 Extension of the Concept of Integral 301a. Introduction. Definition of ImproperIntegrals, 301 b. Functions with InfiniteDiscontinuities, 303 c. Interpretation asAreas ,304 d. Tests for Convergence, 305e Infinite Interval of Integration, 306 f. Theamma Function, 308 g The DirichletIntegral, 309 h. Substitution. FresnelIntegrals, 310

3.16 The Differential Equations of theTrigonometric Functions 312a. Introductory Remarks on DifferentialEquations, 312 b Sin x and cos x defined bya Differential Equation and Initial Conditions,312

PROBLEMS 314

Chapter 4 Applications in Physics and Geometry 3244.1 Theory of Plane Curves 324

a. Parametric Representation, 324 b Changeof Parameters, 326 c. Motion along a Curve.Time as the Parameter. Example of the


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xvi ontentsCycloid, 328 d. Classifications o Curves.Orientation, 333 e Derivatives. Tangent andNormal, in Parametric Representation, 343f The Length o a Curve, 348 g. The ArcLength as a Parameter, 352 h Curvature,354 i Change o Coordinate Axes.Invariance,360 j Uniform Motion in theSpecial Theory o Relativity, 363 k. IntegralsExpressing Area within Closed Curves, 365I Center o Mass and Moment o a Curve, 373m Area and Volume o a Surface oRevolution, 374 D Moment o Inertia, 375

4.2 Examples 376a. The Common Cycloid, 376 b TheCatenary,378 c The Ellipse and theLemniscate, 378

4.3 Vectors in Two Dimensionsa. Definition o Vectors by Translation.Notations, 380 b Addition and Multiplicationo Vectors, 384 c. Variable Vectors, TheirDerivatives, and Integrals, 392 d Applicationto Plane Curves. Direction, Speed, andAcceleration, 394

379

4.4 Motion of a Particle under Given Forces 397a. Newton s Law o Motion, 397 b Motiono Falling Bodies, 398 c Motion o a ParticleConstrained to a Given Curve, 400

4.5 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 o the Motion, 409 d TheOrdinary Pendulum, 410 e. The CycloidalPendulum, 411


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Contents xvii4.8 Motion n a Gravitational Field 413a. Newton s Universal Law of Gravitation, 413

b Circular Motion about the Center ofAttraction, 415 c. Radial Motion EscapeVelocity, 416

4.9 Work and Energy 418a. Work Done by Forces during a Motion, 418b Work and Kinetic Energy. Conservation ofEnergy,420 c The Mutual Attraction ofTwo Masses, 421 d The Stretching of aSpring, 423 e The Charging of a Condenser,423

APPENDIX 424A 1 Properties of the Evolute 424A.2 Areas Bounded by Closed Curves. Indices 430

PROBLEMS 435

Chapter 5 Taylor s Expansion 4405.1 Introduction: Power Series 4405.2 Expansion of the Logarithm and the InverseTangent 442

a. The Logarithm, 442 b The InverseTangent, 444

5.3 Taylor s Theorem 445a. Taylor s Representation of Polynomials, 445b Taylor s Formula for NonpolynomialFunctions, 446

5.4 Expression and Estimates for the Remainder 447a. Cauchy s and Lagrange s Expressions, 447b An Alternative Derivation of Taylor sFormula, 450

5.5 Expansions of the Elementary Functions 453a. The Exponential Function, 453


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xviii Contentsh Expansion of sin x cos x sinh x cosh x 454c The Binomial Series, 456

5.6 Geometrical Applications 457a. Contact of Curves, 458 b On the Theoryof Relative Maxima and Minima, 461

APPENDIX I 462A.I.1 Example of a Function Which Cannot eExpanded in a Taylor Series 462

A.I.2 Zeros and Infinites of Functions 463a. Zeros of Order n 463 b Infinity of Orderv 463

A.I.3 Indeterminate ExpressionsA.I.4 The Convergence of the Taylor Series of aFunction with Nonnegative Derivatives of

464

all Orders 467APPENDIX INTERPOLATION 470A.II.1 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 477

Chapter Numerical Methods 4816 1 Computation of Integrals 482a. Approximation by Rectangles, 482

b Refined Approximations-Simpson s Rule,483


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ontents xix6.2 Other Examples of Numerical Methods 490a. The Calculus of Errors , 490

b. Calculation of 7r 492 c Calculation ofLogarithms, 493

6.3 Numerical Solution of Equations 494a. Newton's Method, 495 b. The Rule of FalsePosition, 497 c. The Method of Iteration, 499d. Iterations and Newton's Procedure, 502

APPENDIXA.l Stirling's FormulaPROBLEMS

hapter Infinite Sums and Products

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5107.1 The Concepts of Convergence and Divergence 511a. Basic Concepts, 511 b Absolute

Convergence and Conditional Convergence, 513c. Rearrangement of Terms, 517d Operations with Infinite Series, 520

7.2 Tests for Absolute Convergence andDivergence 520a. The Comparison Test. Majorants, 520b Convergence Tested by Comparison with theGeometric Series, 521 c Comparison withan Integral, 524

7.3 Sequences of Functions 526a. Limiting Processes with Functions andCurves, 527

7.4 Uniform and Nonuniform Convergence 529a. General Remarks and Definitions, 529b A Test for Uniform Convergence, 534c. Continuity of the Sum of a UniformlyConvergent Series of Continuous Functions, 535d Integration of Uniformly ConvergentSeries, 536 e. Differentiation of InfiniteSeries, 538


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xx Contents7.5 Power Seriesa. Convergence Properties of Power Series-

Interval of Convergence, 540 b Integrationand Differentiation of Power Series, 542c. Operations with Power Series, 543d. Uniqueness of Expansion, 544 e. AnalyticFunctions, 545

7.6 Expansion of Given Functions in Power Series.Method of Undetermined Coefficients.

540

Examples 546a. The Exponential Function, 546 b TheBinomial Series, 546 c. The Series for arcsin x 549 d The Series forar sinh x = log [x yr(1:--+-x-'2 --')), 549e. Example of Multiplication of Series, 550f. Example of Terrn-by-Term IntegrationElliptic Integral), 550

7.7 Power Series with Complex Terms 551a. Introduction of Complex Terms into PowerSeries. Complex Representations of theTrigonometric Function, 551 b A Glance atthe General Theory of Functions of a ComplexVariable, 553

APPENDIX 555A l Multiplication and Division of Series 555a. Multiplication of Absolutely Convergent

Series, 555 b Multiplication and Division ofPower Series, 556A.2 Infinite Series and Improper Integrals 557A.3 Infinite Products 559A.4 Series Involving Bernoulli Numbers 562PROBLEMS 564


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Contents xxi

Chapter 8 Trigonometric Series8.1 Periodic Functionsa. General Remarks. Periodic Extension o a

Function, 572 b. Integrals Over a Period, 573c Harmonic Vibrations, 574

571

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8.2 Superposition of Harmonic Vibrations 576a. Harmonics. Trigonometric Polynomials, 576b Beats, 577

8.3 Complex Notation 582a. General Remarks, 582 b. Application toAlternating Currents, 583 c ComplexNotation for Trigonometrical Polynomials, 585d. A Trigonometric Formula, 586

8.4 Fourier Seriesa. Fourier Coefficients, 587 b. Basic Lemma,588c Proof o dz = - 589co sin z 7ro z 2d Fourier Expansion for theFunction cf x) = x,591 e. The MainTheorem on Fourier Expansion, 593

587

8.5 Examples of Fourier Series 598a. Preliminary Remarks, 598 b Expansion othe Function cf x) = x2, 598 c. Expansiono x cos x, 598 d. TheFunctionf x) = Ix . 600 e A PiecewiseConstant Function, 600 f The Functionsin lxi 601 g. Expansion o cos J l xResolution o the Cotangent into PartialFractions. The Infinite Product for theSine, 602 h Further Examples, 603

8.6 Further Discussion of Convergence 604a. Results, 604 b. Bessel s Inequality, 604


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xxii ontentsc. Proof of Corollaries (a), (b), and (c), 605d. Order of Magnitude of the FourierCoefficients Differentiation of FourierSeries, 607

8 7 Approximation by Trigonometric and RationalPolynomials 608a. General Remark on Representations ofFunctions, 608 b. Weierstrass ApproximationTheorem, 608 c Fejers TrigonometricApproximation of Fourier Polynomials byArithmetical Means, 610 d. Approximationin the Mean and Parseval s Relation, 612

APPENDIX 614

A I l Stretching of the Period Interval. Fourier sIntegral Theorem 614A.I.2 Gibb s Phenomenon at Points ofDiscontinuity 616A.I.3 Integration of Fourier Series 618APPENDIX n 619

A.II.t Bernoulli Polynomials and TheirApplications 619a. Definition and Fourier Expansion, 619b. Generating Functions and the Taylor Seriesof the Trigonometric and HyperbolicCotangent,621 c The Euler-MaclaurinSummation Formula, 624 d. Applications.Asymptotic Expressions, 626 e. Sums ofPower Recursion Formula for BernoulliNumbers,628 f Euler s Constant andStirling s Series, 629

PROBLEMS 631


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Contents xxiii

Chapter Differential Equations or the SimplestTypes o Vibration 69.1 Vibration Problems of Mechanics and Physics 634a. The Simplest Mechanical Vibrations 634

b. Electrical Oscillations 6359.2 Solution of the Homogeneous Equation. FreeOscillations 636a. The Fornal Solution 636 b. Physical

Interpretation o the Solution 638c. Fulfilment o Given Initial Conditions.Uniqueness o the Solution 639

9.3 The Nonhomogeneous Equation. ForcedOscillationsa. General Remarks. Superposition 640b Solution o the NonhomogeneousEquation 642 c The Resonance Curve 643d. Further Discussion o the Oscillation 646e Remarks on the Construction o RecordingInstruments 647

List o Biographical DatesIndex

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IntrotoCalculus(Index)