Introduction to Calculus and Analysis

1 Introduction.- 1.1 The Continuum of Numbers.- a. The System of Natural Numbers and Its Extension. Counting and Measuring, 1 b. Real Numbers and Nested Intervals, 7 c. Decimal Fractions. Bases Other Than Ten, 9 d. Definition of Neighborhood, 12 e. Inequalities, 12.- 1.2 The Concept of Function.- a. Mapping-Graph, 18 b. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function, 21 c. Graphical Representation. Monotonic Functions, 24 d. Continuity, 31 e. The Intermediate Value Theorem. Inverse Functions, 44.- 1.3 The Elementary Functions.- a. Rational Functions, 47 b. Algebraic Functions, 49 c. Trigonometric Functions, 49 d. The Exponential Function and the Logarithm, 51 e. Compound Functions, Symbolic Products, Inverse Functions, 52.- 1.4 Sequences.- 1.5 Mathematical Induction.- 1.6 The Limit of a Sequence.- a. $${a_n} = \frac{1}{n}$$, 61 b. $${a_{2m}} = \frac{1}{m}$$; 62 c. $${a_{2m - 1}} = \frac{1}{{2m}}$$ 63 d. $${a_n} = \sqrt[n]{p}$$, 64 e.an=?n 65 f. Geometrical Illustration of the Limits of ?nand $$\sqrt[n]{p}$$ 65 g. The Geometric Series, 67 h. $${a_n} = \sqrt[n]{n}$$, 67 i. $${a_n} = \sqrt {n + 1} - \sqrt n $$, 69.- 1.7 Further Discussion of the Concept of Limit.- a. Definition of Convergence and Divergence, 70 b. Rational Operations with Limits, 71 c. Intrinsic Convergence Tests. Monotone Sequences, 73 d. Infinite Series and the Summation Symbol, 75 e. The Number e, 77 f. The Number ? as a Limit, 80.- 1.8 The Concept of Limit for Functions of a Continuous Variable.- a. Some Remarks about the Elementary Functions, 86.- Supplements.- S.1 Limits and the Number Concept.- a. The Rational Numbers, 89 b. Real Numbers Determined by Nested Sequences of Rational Intervals, 90 c. Order, Limits, and Arithmetic Operations for Real Numbers, 92 d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria, 94 e. Least Upper Bound and Greatest Lower Bound, 97 f. Denumerability of the Rational Numbers, 98.- S.2 Theorems on Continuous Functions.- S.3 Polar Coordinates.- S.4 Remarks on Complex Numbers.- Problems.- 2 The Fundamental Ideas of the Integral and Differential Calculus.- 2.1 The Integral.- a. Introduction, 120 b. The Integral as an Area, 121 c. Analytic Definition of the Integral. Notations, 122.- 2.2 Elementary Examples of Integration.- a. Integration of Linear Function, 128 b. Integration of x2, 130 c. Integration of x? for Integers ? ? 1, 131 d. Integration of x? for Rational ? Other Than -1, 134 e. Integration of sin x and cos x, 135.- 2.3 Fundamental Rules of Integration.- a. Additivity, 136 b. Integral of a Sum of a Product with a Constant, 137 c. Estimating Integrals, 138, d. The Mean Value Theorem for Integrals, 139.- 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral).- 2.5 Logarithm Defined by an Integral.- a. Definition of the Logarithm Function, 145 b. The Addition Theorem for Logarithms, 147.- 2.6 Exponential Function and Powers.- a. The Logarithm of the Number e, 149 b. The Inverse Function of the Logarithm. The Exponential Function, 150 c. The Exponential Function as Limit of Powers, 152 d. Definition of Arbitrary Powers of Positive Numbers, 152 e. Logarithms to Any Base, 153.- 2.7 The Integral of an Arbitrary Power of x.- 2.8 The Derivative.- a. The Derivative and the Tangent, 156 b. The Derivative as a Velocity, 162 c. Examples of Differentiation, 163 d. Some Fundamental Rules for Differentiation, 165 e. Differentiability and Continuity of Functions, 166 f. Higher Derivatives and Their Significance, 169 g. Derivative and Difference Quotient. Leibnitz's Notation, 171 h. The Mean Value Theorem of Differential Calculus, 173 i. Proof of the Theorem, 175 j. The Approximation of Functions by Linear Functions. Definition of Differentials, 179 k. Remarks on Applications to the Natural Sciences, 183.- 2.9 The Integral, the Primitive Function, and theFundamental Theorems of the Calculus.- a. The Derivative of the Integral, 184 b. The Primitive Function and Its Relation to the Integral, 186 c. The Use of the Primitive Function for Evaluation of Definite Integrals, 189 d. Examples, 191.- Supplement The Existence of the Definite Integral of a Continuous Function.- Problems.- 3 The Techniques of Calculus.- A Differentiation and Integration of the Elementary Functions.- 3.1 The Simplest Rules for Differentiation and Their Applications.- a. Rules for Differentiation, 201 b. Differentiation of the Rational Functions, 204 c. Differentiation of the Trigonometric Functions, 205.- 3.2 The Derivative of the Inverse Function.- a. General Formula, 206 b. The Inverse of the nth Power; the nth Root, 210 c. The Inverse Trigonometric Functions-Multivaluedness, 210 d. The Corresponding Integral Formulas, 215 e. Derivative and Integral of the Exponential Function, 216.- 3.3 Differentiation of Composite Functions.- a. Definitions, 217 b. The Chain Rule, 218 c. The Generalized Mean Value Theorem of the Differential Calculus, 222.- 3.4 Some Applications of the Exponential Function.- a. Definition of the Exponential Function by Means of a Differential Equation, 223 b. Interest Compounded Continuously. Radioactive Disintegration, 224 c. Cooling or Heating of a Body by a Surrounding Medium, 225 d. Variation of the Atmospheric Pressure with the Height above the Surface of the Earth, 226 e. Progress of a Chemical Reaction, 227 f. Switching an Electric Circuit on or off, 228.- 3.5 The Hyperbolic Functions.- a. Analytical Definition, 228 b. Addition Theorems and Formulas for Differentiation 231 c. The Inverse Hyperbolic Functions, 232 d. Further Analogies, 234.- 3.6 Maxima and Minima.- a. Convexity and Concavity of Curves, 236 b. Maxima and Minima-Relative Extrema. Stationary Points, 238.- 3.7 The Order of Magnitude of Functions.- a. The Concept of Order of Magnitude. The Simplest Cases, 248 b. The Order of Magnitude of the Exponential Function and of the Logarithm, 249 c. General Remarks, 251 d. The Order of Magnitude of a Function in the Neighborhood of an Arbitrary Point, 252 e. The Order of Magnitude (or Smallness) of a Function Tending to Zero, 252 f. The "O" and "o" Notation for Orders of Magnitude, 253.- A.1 Some Special Functions.- a. The Function $$y = {e^{1/{x^2}}}$$, 255 b. The Function y = el/x, 256 c. The Function y = tanh 1/x, 257 d. The Function y = x tanh 1/x, 258 e. The Function y = x sin 1/x, y(0) = 0, 259.- A.2 Remarks on the Differentiability of Functions.- B Techniques of Integration.- 3.8 Table of Elementary Integrals.- 3.9 The Method of Substitution.- a. The Substitution Formula. Integral of a Composite Function, 263 b. A Second Derivation of the Substitution Formula, 268 c. Examples. Integration Formulas, 270.- 3.10 Further Examples of the Substitution Method.- 3.11 Integration by Parts.- a. General Formula, 274 b. Further Examples of Integration by Parts, 276 c. Integral Formula for (b) + f(a), 278 d. Recursive Formulas, 278 *e. Wallis's Infinite Product for ?, 280.- 3.12 Integration of Rational Functions.- a. The Fundamental Types, 283 b. Integration of the Fundamental Types, 284 c. Partial Fractions, 286 d. Examples of Resolution into Partial Fractions. Method of Undetermined Coefficients, 288.- 3.13 Integration of Some Other Classes ofFunctions.- a. Preliminary Remarks on the Rational Representation of the Circle and the Hyperbola, 290 b. Integration of R(cos x, sin x), 293 c. Integration of jR(cosh x, sinh x), 294 d. Integration of $$R\left( {x,\sqrt {1 - {x^2}} } \right)$$, 294 e. Integration of $$R\left( {x,\sqrt {{x^2} - 1} } \right)$$, 295 f. Integration of $$R\left( {x,\sqrt {{x^2} + 1} } \right)$$, 295 g. Integration of $$R\left( {x,\sqrt {a{x^2} + 2bx + c} } \right)$$, 295 h. Further Examples of Reduction to Integrals of Rational Functions, 296 i. Remarks on the Examples, 297.- C Further Steps in the Theory of Integral Calculus.- 3.14 Integrals of Elementary Functions.- a. Definition of Functions by Integrals. Elliptic Integrals and Functions, 298 b. On Differentiation and Integration, 300.- 3.15 Extension of the Concept of Integral.- a. Introduction. Definition of "Improper" Integrals, 301 b. Functions with Infinite Discontinuities, 303 c. Interpretation as Areas, 304 d. Tests for Convergence, 305 e. Infinite Interval of Integration, 306 f. The Gamma Function, 308 g. The Dirichlet Integral, 309 h. Substitution. Fresnel Integrals, 310.- 3.16 The Differential Equations of the Trigonometric Functions.- a. Introductory Remarks on Differential Equations, 312 b. Sin x and cos x defined by a Differential Equation and Initial Conditions, 312.- Problems.- 4 Applications in Physics and Geometry.- 4.1 Theory of Plane Curves.- a. Parametric Representation, 324 b. Change of Parameters, 326 c. Motion along a Curve. Time as the Parameter. Example of the Cycloid, 328 d. Classifications of Curves. Orientation, 333 e. Derivatives. Tangent and Normal, in Parametric Representation, 343 f. The Length of a Curve, 348 g. The Arc Length as a Parameter, 352 h. Curvature, 354 i. Change of Coordinate Axes. Invariance, 360 j. Uniform Motion in the Special Theory of Relativity, 363 k. Integrals Expressing Area within Closed Curves, 365 1. Center of Mass and Moment of a Curve, 373 m. Area and Volume of a Surface of Revolution, 374 n. Moment of Inertia, 375.- 4.2 Examples.- a. The Common Cycloid, 376 b. The Catenary, 378 c. The Ellipse and the Lemniscate, 378.- 4.3 Vectors in Two Dimensions.- a. Definition of Vectors by Translation. Notations, 380 b. Addition and Multiplication of Vectors, 384 c. Variable Vectors, Their Derivatives, and Integrals, 392 d. Application to Plane Curves. Direction, Speed, and Acceleration, 394.- 4.4 Motion of a Particle under Given Forces.- a. Newton's Law of Motion, 397 b. Motion of Falling Bodies, 398 c. Motion of a Particle Constrained to a Given Curve, 400.- 4.5 Free Fall of a Body Resisted by Air.- 4.6 The Simplest Type of Elastic Vibration.- 4.7 Motion on a Given Curve.- a. The Differential Equation and Its Solution, 405 b. Particle Sliding down a Curve, 407 c. Discussion of the Motion, 409 d. The Ordinary Pendulum, 410 e. The Cycloidal Pendulum, 411.- 4.8 Motion in a Gravitational Field.- a. Newton's Universal Law of Gravitation, 413 b. Circular Motion about the Center of Attraction, 415 c. Radial Motion-Escape Velocity, 416.- 4.9 Work and Energy.- a. Work Done by Forces during a Motion, 418 b. Work and Kinetic Energy. Conservation of Energy, 420 c. The Mutual Attraction of Two Masses, 421 d. The Stretching of a Spring, 423 e. The Charging of a Condenser, 423.- A.1 Properties of the Evolute.- A.2 Areas Bounded by Closed Curves. Indices.- Problems.- 5 Taylor's Expansion.- 5.1 Introduction: Power Series.- 5.2 Expansion of the Logarithm and the Inverse Tangent.- a. The Logarithm, 442 b. The Inverse Tangent, 444.- 5.3 Taylor's Theorem.- a. Taylor's Representation of Polynomials, 445 b. Taylor's Formula for Nonpolynomial Functions, 446.- 5.4 Expression and Estimates for the Remainder.- a. Cauchy's and Lagrange's Expressions, 447 b. An Alternative Derivation of Taylor's Formula, 450.- 5.5 Expansions of the Elementary Functions.- a. The Exponential Function, 453 b. Expansion of sin x cos x, sinh x cosh x, 454 c. The Binomial Series, 456.- 5.6 Geometrical Applications.- a. Contact of Curves, 458 b. On the Theory of Relative Maxima and Minima, 461.- Appendix I.- A.I.1 Example of a Function Which Cannot Be Expanded in a Taylor Series.- A.I.2 Zeros and Infinites of Functions.- a. Zeros of Order n, 463 b. Infinity of Order v, 463.- A.I.3 Indeterminate Expressions.- A.I.4 The Convergence of the Taylor Series of a Function with Nonnegative Derivatives of all Orders.- Appendix II Interpolation.- A.II.1 The Problem of Interpolation. Uniqueness.- A.II.2 Construction of the Solution. Newton's Interpolation Formula.- A.II.3 The Estimate of the Remainder.- A.II.4 The Lagrange Interpolation Formula.- Problems.- 6 Numerical Methods.- 6.1 Computation of Integrals.- a. Approximation by Rectangles, 482 b. Refined Approximations-Simpson's Rule, 483.- 6.2 Other Examples of Numerical Methods.- a. The "Calculus of Errors", 490 b. Calculation of ?, 492 c. Calculation of Logarithms, 493.- 6.3 Numerical Solution of Equations.- a. Newton's Method, 495 b. The Rule of False Position, 497 c. The Method of Iteration, 499 d. Iterations and Newton's Procedure, 502.- A.1 Stirling's Formula.- Problems.- 7 Infinite Sums and Products.- 7.1 The Concepts of Convergence and Divergence.- a. Basic Concepts, 511 b. Absolute Convergence and Conditional Convergence, 513 c. Rearrangement of Terms, 517 d. Operations with Infinite Series, 520.- 7.2 Tests for Absolute Convergence and Divergence.- a. The Comparison Test. Majorants, 520 b. Convergence Tested by Comparison with the Geometric Series, 521 c. Comparison with an Integral, 524.- 7.3 Sequences of Functions.- a. Limiting Processes with Functions and Curves, 527.- 7.4 Uniform and Nonuniform Convergence.- a. General Remarks and Definitions, 529 b. A Test for Uniform Convergence, 534 c. Continuity of the Sum of a Uniformly Convergent Series of Continuous Functions, 535 d. Integration of Uniformly Convergent Series, 536 e. Differentiation of Infinite Series, 538.- 7.5 Power Series.- a. Convergence Properties of Power Series-Interval of Convergence, 540 b. Integration and Differentiation of Power Series, 542 c. Operations with Power Series, 543 d. Uniqueness of Expansion, 544 e. Analytic Functions, 545.- 7.6 Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples.- a. The Exponential Function, 546 b. The Binomial Series, 546 c. The Series for arc sin x, 549 d. The Series for ar sinh $$x = \log \left[ {x + \sqrt {\left( {1 + {x^2}} \right)} } \right]$$, 549 e. Example of Multiplication of Series, 550 f. Example of Term-by-Term Integration (Elliptic Integral), 550.- 7.7 Power Series with Complex Terms.- a. Introduction of Complex Terms into Power Series. Complex Representations of the Trigonometric Function, 551 b. A Glance at the General Theory of Functions of a Complex Variable, 553.- A.1 Multiplication and Division of Series.- a. Multiplication of Absolutely Convergent Series, 555 b. Multiplication and Division of Power Series, 556.- A.2 Infinite Series and Improper Integrals.- A.3 Infinite Products.- A.4 Series Involving Bernoulli Numbers.- Problems.- 8 Trigonometrie Series.- 8.1 Periodic Functions.- a. General Remarks. Periodic Extension of a Function, 572 b. Integrals Over a Period, 573 c. Harmonie Vibrations, 574.- 8.2 Superposition of Harmonic Vibrations.- a. Harmonics. Trigonometric Polynomials, 576 b. Beats, 577.- 8.3 Complex Notation.- a. General Remarks, 582 b. Application to Alternating Currents, 583 c. Complex Notation for Trigonometrical Polynomials, 585 d. A Trigonometric Formula, 586.- 8.4 Fourier Series.- a. Fourier Coefficients, 587 b. Basic Lemma, 588 c. Proof of $$\int_0^\infty {\frac{{\sin z}}{z}dz = \frac{\pi }{2}} $$, 589 d. Fourier Expansion for the Function ø (x)= x 591 e. The Main Theorem on Fourier Expansion, 593.- 8.5 Examples of Fourier Series.- a. Preliminary Remarks, 598 b. Expansion of the Function ø (x) = x2, 598 c. Expansion of x cos x, 598 d. The Function f(x) = |x|, 600 e. A Piecewise Constant Function, 600 f. The Function sin |x|, 601 g. Expansion of cos µx. Resolution of the Cotangent into Partial Fractions. The Infinite Product for the Sine, 602 h. Further Examples, 603.- 8.6 Further Discussion of Convergence.- a. Results, 604 b. BesseFs Inequality, 604 c. Proof of Corollaries (a), (b), and (c), 605 d. Order of Magnitude of the Fourier Coefficients Differentiation of Fourier Series, 607.- 8.7 Approximation by Trigonometric and RationalPolynomials.- a. General Remark on Representations of Functions, 608 b. Weierstrass Approximation Theorem, 608 c. Fejers Trigonometric Approximation of Fourier Polynomials by Arithmetical Means, 610 d. Approximation in the Mean and Parseval's Relation, 612.- Appendix I.- A.I.1 Stretching of the Period Interval. Fourier's Integral Theorem.- A.I.2 Gibb's Phenomenon at Points of Discontinuity.- A.I.3 Integration of Fourier Series.- Appendix II.- A.II.1 Bernoulli Polynomials and Their Applications.- a. Definition and Fourier Expansion, 619 b. Generating Functions and the Taylor Series of the Trigonometric and Hyperbolic Cotangent, 621 c. The Euler-Maclaurin Summation Formula, 624 d. Applications. Asymptotic Expressions, 626 e. Sums of Power Recursion Formula for Bernoulli Numbers, 628 f. Euler's Constant and Stirling's Series, 629.- Problems.- 9 Differential Equations jor the SimplestTypes of Vibration.- 9.1 Vibration Problems of Mechanics and Physics.- a. The Simplest Mechanical Vibrations, 634 b. Electrical Oscillations, 635.- 9.2 Solution of the Homogeneous Equation. Free Oscillations.- a. The Fomai Solution, 636 b. Physical Interpretation of the Solution, 638 c. Fulfilment of Given Initial Conditions. Uniqueness of the Solution, 639.- 9.3 The Nonhomogeneous Equation. Forced Oscillations.- a. General Remarks. Superposition, 640 b. Solution of the Nonhomogeneous Equation, 642 c. The Resonance Curve, 643 d. Further Discussion of the Oscillation, 646 e. Remarks on the Construction of Recording Instruments, 647.- List of Biographical Dates.