## Analysis by Its History. . . that departed from the traditional dry-as-dust mathematics textbook. (M. Kline, from the Preface to the paperback edition of Kline 1972) Also for this reason, I have taken the trouble to make a great number of drawings. (Brieskom & Knorrer, Plane algebraic curves, p. ii) . . . I should like to bring up again for emphasis . . . points, in which my exposition differs especially from the customary presentation in the text books: 1. Illustration of abstract considerations by means of figures. 2. Emphasis upon its relation to neighboring fields, such as calculus of dif ferences and interpolation . . . 3. Emphasis upon historical growth. It seems to me extremely important that precisely the prospective teacher should take account of all of these. (F. Klein 1908, Eng\. ed. p. 236) Traditionally, a rigorous first course in Analysis progresses (more or less) in the following order: limits, sets, '* continuous '* derivatives '* integration. mappings functions On the other hand, the historical development of these subjects occurred in reverse order: Archimedes Cantor 1875 Cauchy 1821 Newton 1665 . ;::: Kepler 1615 Dedekind . ;::: Weierstrass . ;::: Leibniz 1675 Fermat 1638 In this book, with the four chapters Chapter I. Introduction to Analysis of the Infinite Chapter II. Differential and Integral Calculus Chapter III. Foundations of Classical Analysis Chapter IV. Calculus in Several Variables, we attempt to restore the historical order, and begin in Chapter I with Cardano, Descartes, Newton, and Euler's famous Introductio. |

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### Contents

I1 Cartesian Coordinates and Polynomial Functions | 2 |

Algebra Nova | 6 |

Descartess Geometry | 8 |

Polynomial Functions | 10 |

Exercises | 14 |

I2 Exponentials and the Binomial Theorem | 17 |

Binomial Theorem | 18 |

Exponential Function | 25 |

Criteria for Convergence | 189 |

Absolute Convergence | 192 |

Double Series | 195 |

The Cauchy Product of Two Series | 197 |

Exchange of Inﬁnite Series and Limits | 199 |

Exercises | 200 |

III3 Real Functions and Continuity | 202 |

Continuous Functions | 204 |

Exercises | 28 |

I3 Logarithms and Areas | 29 |

Computation of Logarithms | 30 |

Computation of Areas | 33 |

Area of the Hyperbola and Natural Logarithms | 34 |

Exercises | 39 |

I4 Trigonometric Functions | 40 |

Basic Relations and Consequences | 43 |

Series Expansions | 46 |

Inverse Trigonometric Functions | 49 |

Computation of Pi | 52 |

Exercises | 55 |

I5 Complex Numbers and Functions | 57 |

Eulers Formula and Its Consequences | 58 |

A New View on Trigonometric Functions | 61 |

Eulers Product for the Sine Function | 62 |

Exercises | 66 |

I6 Continued Fractions | 68 |

Convergents | 71 |

Irrationality | 76 |

Exercises | 78 |

Differential and Integral Calculus | 80 |

II1 The Derivative | 81 |

Differentiation Rules | 84 |

Parametric Representation and Implicit Equations | 88 |

Exercises | 89 |

II2 Higher Derivatives and Taylor Series | 91 |

De Conversione Functionum in Series | 94 |

Exercises | 97 |

II3 Envelopes and Curvature | 98 |

The Caustic of a Circle | 99 |

Envelope of Ballistic Curves | 101 |

Exercises | 105 |

II4 Integral Calculus | 107 |

Applications | 109 |

Integration Techniques | 112 |

Taylors Formula with Remainder | 116 |

Exercises | 117 |

II5 Functions with Elementary Integral | 118 |

Useful Substitutions | 123 |

Exercises | 125 |

II6 Approximate Computation of Integrals | 126 |

Numerical Methods | 128 |

Asymptotic Expansions | 131 |

Exercises | 132 |

II7 Ordinary Differential Equations | 134 |

Some Types of Integrable Equations | 139 |

SecondOrder Differential Equations | 140 |

Exercises | 143 |

II8 Linear Differential Equations | 144 |

Homogeneous Equation with Constant Coefficients | 145 |

Inhomogeneous Linear Equations | 148 |

Cauchys Equation | 152 |

II9 Numerical Solution of Differential Equations | 154 |

Taylor Series Method | 156 |

SecondOrder Equations | 158 |

Exercises | 159 |

1110 The EulerMaclaurin Summation Formula | 160 |

De Usu Legitimo Formulae Summatoriae Maclaurinianae | 163 |

Stirlings Formula | 165 |

The Harmonic Series and Eulers Constant | 167 |

Exercises | 169 |

Foundations of Classical Analysis | 170 |

III1 Infinite Sequences and Real Numbers | 172 |

Construction of Real Numbers | 177 |

Monotone Sequences and Least Upper Bound | 182 |

Accumulation Points | 184 |

Exercises | 185 |

III2 Infinite Series | 188 |

The Intermediate Value Theorem | 206 |

Monotone and Inverse Functions | 208 |

Limit of a Function | 209 |

Exercises | 210 |

III4 Uniform Convergence and Uniform Continuity | 213 |

Weierstrasss Criterion for Uniform Convergence | 216 |

Uniform Continuity | 217 |

Exercises | 220 |

III5 The Riemann Integral | 221 |

Integrable Functions | 226 |

Inequalities and the Mean Value Theorem | 228 |

Integration of Inﬁnite Series | 230 |

Exercises | 232 |

III6 Differentiable Functions | 235 |

The Fundamental Theorem of Differential Calculus | 239 |

The Rules of de LHospital | 242 |

Derivatives of Infinite Series | 245 |

Exercises | 246 |

III7 Power Series and Taylor Series | 248 |

Determination of the Radius of Convergence | 249 |

Continuity | 250 |

Differentiation and Integration | 251 |

Taylor Series | 252 |

Exercises | 255 |

III 8 Improper Integrals | 257 |

Unbounded Functions on a Finite Interval | 260 |

Eulers Gamma Function | 261 |

Exercises | 262 |

III9 Two Theorems on Continuous Functions | 263 |

Weierstrasss Approximation Theorem | 265 |

Exercises | 269 |

Calculus in Several Variables | 271 |

IV1 Topology of nDimensional Space | 273 |

Convergence of Vector Sequences | 275 |

Neighborhoods Open and Closed Sets | 278 |

Compact Sets | 283 |

Exercises | 285 |

IV2 Continuous Functions | 287 |

Continuous Functions and Compactness | 289 |

Uniform Continuity and Uniform Convergence | 290 |

Linear Mappings | 293 |

Hausdorffs Characterization of Continuous Functions | 294 |

Integrals with Parameters | 297 |

Exercises | 298 |

IV3 Differentiable Functions of Several Variables | 300 |

Differentiability | 302 |

Counterexamples | 304 |

A Geometrical Interpretation of the Gradient | 305 |

The Mean Value Theorem | 308 |

The Implicit Function Theorem | 309 |

Differentiation of Integrals with Respect to Parameters | 311 |

Exercises | 313 |

IV4 Higher Derivatives and Taylor Series | 316 |

Taylor Series for Two Variables | 319 |

Taylor Series for n Variables | 320 |

Maximum and Minimum Problems | 323 |

Conditional Minimum Lagrange Multiplier | 325 |

Exercises | 328 |

IV5 Multiple Integrals | 330 |

Null Sets and Discontinuous Functions | 334 |

Arbitrary Bounded Domains | 336 |

The Transformation Formula for Double Integrals | 338 |

Integrals with Unbounded Domain | 345 |

Exercises | 347 |

Original Quotations | 351 |

References | 358 |

Symbol Index | 369 |

Index | 371 |

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### Common terms and phrases

absolutely convergent accumulation point Algebra apply arbitrary arctan Bernoulli bounded calculus Cantor Cauchy product Cauchy sequence coefﬁcients complex number compute consider the function constant continuous function cosx counterexample curve Darboux Darboux sums deﬁned Deﬁnition denote differential equation diverges Euler example Exercises exists exponential function ﬁnd ﬁrst fn(x formula fraction function f(x Gen`eve geometric given Hence inﬁnite series integral interval inverse inverse function Joh.Bernoulli L’Hospital Lagrange Leibniz Lemma Let f limit linear logarithms Math mathematics method neighborhood Newton norm null set obtain Oeuvres Opera partial derivatives polynomial problem Proof prove radius radius of convergence real number Reproduced with permission Riemann roots satisfying Sect sequence sn Show sinx solution tangent tanx Taylor series Theorem triangle inequality uniform continuity uniform convergence variables vector Weierstrass Werke

### References to this book

Mathematics Teaching Practice: Guide for University and College Lecturers John Mason No preview available - 2002 |