# Analysis by Its History

Springer Science & Business Media, Jun 2, 2008 - Mathematics - 382 pages
. . . 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.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### 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 Copyright