The higher calculus. A history of real and complex analysis from Euler to Weierstrass

Cover page......Page 1
Title page......Page 2
Contents......Page 6
Introduction......Page 8
1.1 Euler's Concept of a Function......Page 14
1.2 Working with "Imaginary" Quantities......Page 19
1.3 The Debate over Vibrating Strings......Page 28
1.4 Changes in the Concept of a Function......Page 40
Notes to Chapter 1......Page 45
2.1 Mathematics in France after the Revolution......Page 51
2.2 The "Algebraicization" of Analysis......Page 55
2.3 Fourier Series......Page 64
2.3.a. Fourier's Equation for Heat Diffusion......Page 66
2.3.b. The Introduction of Trigonometric Series......Page 74
2.4 Fourier's 'Program'......Page 85
Notes to Chapter 2......Page 88
3.1 Problems with the Foundations of Analysis......Page 92
3.2 Gauss' "Geometrical Rigor"......Page 99
3.3 Bernhard Bolzano......Page 103
3.4 Cauchy's Cours d'analyse......Page 108
a) Limits and Continuity......Page 110
b) The Convergence of Series......Page 115
3.5 Abel's Theorems on Convergence......Page 120
3.6 Cauchy's Definitions of Derivative and Differential......Page 124
Notes to Chapter 3......Page 129
4.1 The Fundamental Theorem of Algebra......Page 133
4.2 Cauchy's Memoire on Definite Integrals......Page 139
4.3 Complex Numbers and Complex Variables......Page 145
4.4 Cauchy's Theory of Integration......Page 149
a) The Definite Integral......Page 150
b) The Integration of Ordinary Differential Equations......Page 153
c) Complex Integration......Page 158
4.5 The "Calcul des limites" and Differential Equations......Page 164
4.6 The Emergence of Cauchy's Theory of Complex Functions......Page 169
Notes to Chapter 4......Page 181
5.1 The "Demonstrations" of Cauchy and Poisson......Page 190
5.2 Lejeune-Dirichlet's Memoir......Page 197
5.3 Dirichlet's Concept of a Function......Page 203
5.4 The Uniform Convergence of Series......Page 209
Notes to Chapter 4......Page 215
6.1 The Character of Bernhard Riemann's Works......Page 220
6.2 The Foundation of the Theory of Complex Functions......Page 222
a) Riemann Surfaces......Page 226
b) Dirichlet's Principle......Page 236
c) A Glance Ahead......Page 241
6.3 Research on Integration and Trigonometric Series......Page 247
Notes to Chapter 6......Page 257
7.1 Discussions with Kronecker and Weierstrass......Page 264
a) 'Man Arithmetizes'......Page 272
b) Trigonometric Series and the Theory of Sets......Page 278
7.3 Weierstrass' Theory of Functions......Page 287
Notes to Chapter 7......Page 298
Appendix: On the History of "Dirichlet's Principle"......Page 302
Notes to the Appendix......Page 310
Bibliography......Page 311
Index......Page 332