A history of analysis / Hans Niels Jahnke, editor.

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Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking. Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems.This volume is designed as a collective work of authors who are proven experts in the history of mathematics. It clarifies the conceptual change that analysis underwent during its development while elucidating the influence of specific applications and describing the relevance of biographical and philosophical backgrounds. The first ten chapters of the book outline chronological development and the last three chapters survey the history of differential equations, the calculus of variations, and functional analysis. Special features are a separate chapter on the development of the theory of complex functions in the nineteenth century and two chapters on the influence of physics on analysis.One is about the origins of analytical mechanics, and one treats the development of boundary-value problems of mathematical physics (especially potential theory) in the nineteenth century. The book presents an accurate and very readable account of the history of analysis. Each chapter provides a comprehensive bibliography. Mathematical examples have been carefully chosen so that readers with a modest background in mathematics can follow them. It is suitable for mathematical historians and a general mathematical audience.

Innehållsförteckning levererad av Syndetics

• Introduction (p. vii)
• Chapter 1. Antiquity (p. 1)
• 1.1. Greek mathematics' part in the formation of analysis (p. 1)
• 1.2. The Greek concept of numbers and magnitudes (p. 3)
• 1.3. Problems of quadrature. An example: The circle (p. 14)
• 1.4. Archimedes' contributions to infinitestimal mathematics (p. 21)
• 1.5. The concept of curves in antiquity (p. 29)
• 1.6. Philosophic reflections on the infinitestimal (p. 32)
• Bibliography (p. 36)
• Chapter 2. Precursors of Differentiation and Integration (p. 41)
• 2.1. Scope and motivation (p. 41)
• 2.2. The study of curves in the 1659 edition of Geometria (p. 42)
• 2.3. Early integration, reflected in the correspondence of Huygens and Sluse (1658) (p. 56)
• 2.4. Barrow glimpses the "fundamental theorem" (p. 69)
• Bibliography (p. 71)
• Chapter 3. Newton's Method and Leibniz's Calculus (p. 73)
• 3.1. Introduction (p. 73)
• 3.2. Newton's method of series and fluxions (p. 74)
• 3.3. Leibniz's differential and integral calculus (p. 85)
• 3.4. Mathematizing force (p. 91)
• 3.5. Newton versus Leibniz (p. 95)
• Bibliography (p. 102)
• Chapter 4. Algebraic Analysis in the 18th Century (p. 105)
• 4.1. Concepts, problems, characters (p. 105)
• 4.2. The example of the catenary (p. 108)
• 4.3. Taylor's theorem (p. 111)
• 4.4. The notion of analytical function (p. 113)
• 4.5. Calculating with series (p. 118)
• 4.6. Limitations of the analytic function concept (p. 123)
• 4.7. Lagrange's algebraic foundation of analysis (p. 127)
• 4.8. The generality of algebra (p. 131)
• Bibliography (p. 132)
• Chapter 5. The Origins of Analytic Mechanics in the 18th Century (p. 137)
• 5.1. The principle of least action: Maupertuis, Euler and Lagrange (1740-1761) (p. 138)
• 5.2. The analytical mechanics (p. 147)
• Bibliography (p. 152)
• Chapter 6. The Foundation of Analysis in the 19th Century (p. 155)
• 6.1. Introduction (p. 155)
• 6.2. The concept of function (p. 156)
• 6.3. Cauchy and the Cours d'analyse (p. 160)
• 6.4. Gauss, Bolzano and Abel (p. 173)
• 6.5. Convergence of Fourier series (p. 178)
• 6.6. Cauchy's theorem and uniform convergence (p. 181)
• 6.7. Weierstrass (p. 184)
• 6.8. Pathological functions and the new style in analysis (p. 187)
• 6.9. Diffusion and acceptance of rigourist analysis (p. 188)
• 6.10. Breaking the rigorous chains (p. 190)
• Bibliography (p. 191)
• Chapter 7. Analysis and Physics in the Nineteenth Century: The Case of Boundary-value Problems (p. 197)
• 7.1. Introduction: Mathematical analysis in physics circa 1800 (p. 197)
• 7.2. Green, Gauss, Dirichlet: Boundary-value problems come of age (p. 202)
• 7.3. Some later developments (p. 209)
• 7.4. Concluding remarks (p. 211)
• Bibliography (p. 211)
• Chapter 8. Complex Function Theory, 1780-1900 (p. 213)
• 8.1. Introduction (p. 213)
• 8.2. "The passage from the real to the imaginary" (p. 214)
• 8.3. "Complex functions and the integral theorem" (p. 219)
• 8.4. The integral formula and the "calcul des limites" (p. 225)
• 8.5. The emergence of the French "school" (p. 227)
• 8.6. Riemann's complex function theory (p. 232)
• 8.7. Riemann's further research (p. 238)
• 8.8. The influence of Riemann's ideas (p. 244)
• 8.9. Weierstrass's early papers (p. 247)
• 8.10. Weierstrass's Funktionenlehre (p. 250)
• Bibliography (p. 255)
• Chapter 9. Theory of Measure and Integration from Riemann to Lebesgue (p. 261)
• 9.1. On the prehistory of Riemann's integral (p. 261)
• 9.2. The Riemann integral (p. 263)
• 9.3. Discussions (p. 266)
• 9.4. Integration revisited: Camille Jordan (p. 275)
• 9.5. The development of the theory of measure (p. 278)
• 9.6. Seeking new directions: Henri Lebesgue (p. 284)
• Bibliography (p. 289)
• Chapter 10. The End of the Science of Quantity: Foundations of Analysis, 1860-1910 (p. 291)
• 10.1. Constructions of real numbers (p. 292)
• 10.2. The emergence of set theory (p. 304)
• 10.3. The axiomatic method (p. 313)
• Bibliography (p. 321)
• Chapter 11. Differential Equations: A Historical Overview to circa 1900 (p. 325)
• 11.1. Introduction (p. 325)
• 11.2. From the origins of calculus to the late eighteenth century (p. 326)
• 11.3. From the French Revolution to about 1900 (p. 339)
• Bibliography (p. 351)
• Chapter 12. The Calculus of Variations: A Historical Survey (p. 355)
• 12.1. Introduction (p. 355)
• 12.2. Prehistory (p. 356)
• 12.3. The Bernoullis, Taylor and Euler (p. 357)
• 12.4. Lagrange (p. 361)
• 12.5. Legendre (p. 363)
• 12.6. Jacobi (p. 364)
• 12.7. Mayer (p. 367)
• 12.8. Erdmann (p. 368)
• 12.9. Weierstrass (p. 370)
• 12.10. Refinement of Weierstrass's methods (p. 374)
• 12.11. Variational methods in mechanics (p. 377)
• 12.12. Existence questions (p. 379)
• Bibliography (p. 380)
• Chapter 13. The Origins of Functional Analysis (p. 385)
• 13.1. Introduction: Summary plus mathematical and historiographical motivations (p. 385)
• 13.2. The roots in the calculus of variations and in the Italian Calcolo Funzionale (p. 387)
• 13.3. Ascoli's theorem, the set-theoretic impulse and Frechet's analyse generale (p. 389)
• 13.4. The roots in the theory of systems of linear equations and integral equations (p. 391)
• 13.5. Pioneering forays without effect: Axiom systems of Peano and Pincherle for infinite-dimensional vector spaces (p. 394)
• 13.6. The Hilbert theory of integral equations and its reformulation (p. 394)
• 13.7. The failed attempt at a synthesis by an outsider: E. H. Moore's General Analysis (p. 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 (p. 398)
• 13.9. Further development of the theory of functionals: Representation theorems (p. 400)
• 13.10. Riesz and the beginning of operator theory (p. 401)
• 13.11. Banach and the Polish school (p. 402)
• 13.12. Conclusion (p. 403)
• Bibliography (p. 405)
• Index of Names (p. 409)
• Subject Index (p. 415)