A history of analysis / Hans Niels Jahnke, editor.

Format:

Book

Contributor:

Jahnke, H. N. (Hans Niels), 1948-

Series:

History of mathematics 0899-2428 ; v. 24.

History of mathematics, 0899-2428 ; v. 24

Language:

English

Subjects (All):

Mathematical analysis--History.

Mathematical analysis.

History.

Physical Description:

ix, 422 pages : illustrations ; 26 cm.

Place of Publication:

[Providence, RI] : American Mathematical Society ; [London] : London Mathematical Society, [2003]

Contents:

Chapter 1. Antiquity / Rudiger Thiele 1

1.1. Greek mathematics' part in the formation of analysis 1

1.2. The Greek concept of numbers and magnitudes 3

1.3. Problems of quadrature. An example: The circle 14

1.4. Archimedes' contributions to infinitestimal mathematics 21

1.5. The concept of curves in antiquity 29

1.6. Philosophic reflections on the infinitestimal 32

Chapter 2. Precursors of Differentiation and Integration / Jan van Maanen 41

2.2. The study of curves in the 1659 edition of Geometria 42

2.3. Early integration, reflected in the correspondence of Huygens and Sluse (1658) 56

2.4. Barrow glimpses the "fundamental theorem" 69

Chapter 3. Newton's Method and Leibniz's Calculus / Niccolo Guicciardini 73

3.2. Newton's method of series and fluxions 74

3.3. Leibniz's differential and integral calculus 85

3.4. Mathematizing force 91

3.5. Newton versus Leibniz 95

Chapter 4. Algebraic Analysis in the 18th Century / Hans Niels Jahnke 105

4.1. Concepts, problems, characters 105

4.2. The example of the catenary 108

4.3. Taylor's theorem 111

4.4. The notion of analytical function 113

4.5. Calculating with series 118

4.6. Limitations of the analytic function concept 123

4.7. Lagrange's algebraic foundation of analysis 127

4.8. The generality of algebra 131

Chapter 5. The Origins of Analytic Mechanics in the 18th Century / Marco Panza 137

5.1. The principle of least action: Maupertuis, Euler and Lagrange (1740-1761) 138

5.2. The analytical mechanics 147

Chapter 6. The Foundation of Analysis in the 19th Century / Jesper Lutzen 155

6.2. The concept of function 156

6.3. Cauchy and the Cours d'analyse 160

6.4. Gauss, Bolzano and Abel 173

6.5. Convergence of Fourier series 178

6.6. Cauchy's theorem and uniform convergence 181

6.7. Weierstrass 184

6.8. Pathological functions and the new style in analysis 187

6.9. Diffusion and acceptance of rigourist analysis 188

6.10. Breaking the rigorous chains 190

Chapter 7. Analysis and Physics in the Nineteenth Century: The Case of Boundary-value Problems / Tom Archibald 197

7.1. Introduction: Mathematical analysis in physics circa 1800 197

7.2. Green, Gauss, Dirichlet: Boundary-value problems come of age 202

7.3. Some later developments 209

Chapter 8. Complex Function Theory, 1780-1900 / Umberto Bottazzini 213

8.2. "The passage from the real to the imaginary" 214

8.3. "Complex functions and the integral theorem" 219

8.4. The integral formula and the "calcul des limites" 225

8.5. The emergence of the French "school" 227

8.6. Riemann's complex function theory 232

8.7. Riemann's further research 238

8.8. The influence of Riemann's ideas 244

8.9. Weierstrass's early papers 247

8.10. Weierstrass's Funktionenlehre 250

Chapter 9. Theory of Measure and Integration from Riemann to Lebesgue / Thomas Hochkirchen 261

9.1. On the prehistory of Riemann's integral 261

9.2. The Riemann integral 263

9.4. Integration revisited: Camille Jordan 275

9.5. The development of the theory of measure 278

9.6. Seeking new directions: Henri Lebesgue 284

Chapter 10. The End of the Science of Quantity: Foundations of Analysis, 1860-1910 / Moritz Epple 291

10.1. Constructions of real numbers 292

10.2. The emergence of set theory 304

10.3. The axiomatic method 313

Chapter 11. Differential Equations: A Historical Overview to circa 1900 / Tom Archibald 325

11.2. From the origins of calculus to the late eighteenth century 326

11.3. From the French Revolution to about 1900 339

Chapter 12. The Calculus of Variations: A Historical Survey / Craig Fraser 355

12.2. Prehistory 356

12.3. The Bernoullis, Taylor and Euler 357

12.4. Lagrange 361

12.5. Legendre 363

12.6. Jacobi 364

12.7. Mayer 367

12.8. Erdmann 368

12.9. Weierstrass 370

12.10. Refinement of Weierstrass's methods 374

12.11. Variational methods in mechanics 377

12.12. Existence questions 379

Chapter 13. The Origins of Functional Analysis / Reinhard Siegmund-Schultze 385

13.1. Introduction: Summary plus mathematical and historiographical motivations 385

13.2. The roots in the calculus of variations and in the Italian Calcolo Funzionale 387

13.3. Ascoli's theorem, the set-theoretic impulse and Frechet's analyse generale 389

13.4. The roots in the theory of systems of linear equations and integral equations 391

13.5. Pioneering forays without effect: Axiom systems of Peano and Pincherle for infinite-dimensional vector spaces 394

13.6. The Hilbert theory of integral equations and its reformulation / E. Schmidt 394

13.7. The failed attempt at a synthesis by an outsider: E. H. Moore's General Analysis 397

13.8. A first synthesis of Frechet's analyse generale and Hilbert-Schmidt's theory of integral equations: The theorem of Riesz and Fischer 398

13.9. Further development of the theory of functionals: Representation theorems 400

13.10. Riesz and the beginning of operator theory 401

13.11. Banach and the Polish school 402.

Notes:

Includes bibliographical references and indexes.

ISBN:

0821826239

OCLC:

51607350