An understanding of the developments in classical analysis during the nineteenth century is vital to a full appreciation of the history of twentieth-century mathematical thought. It was during the nineteenth century that the diverse mathematical formulae of the eighteenth century were systematized and the properties of functions of real and complex variables clearly distinguished; and it was then that the calculus matured into the rigorous discipline of today, becoming in the process a dominant influence on mathematics and mathematical physics. This Source Book, a sequel to D. J. Struik's Source Book in Mathematics, 1200-1800, draws together more than eighty selections from the writings of the most influential mathematicians of the period. Thirteen chapters, each with an introduction by the editor, highlight the major developments in mathematical thinking over the century. All material is in English, and great care has been taken to maintain a high standard of accuracy both in translation and in transcription. Of particular value to historians and philosophers of science, the Source Book should serve as a vital reference to anyone seeking to understand the roots of twentieth-century mathematical thought.
Mr. Birkhoff is the author of a number of well-known books on mathematics and its applications, and is George Putnam Professor of Pure and Applied Mathematics at Harvard University.
PART 1: FOUNDATIONS OF REAL ANALYSIS A. Cauchy's Partial Rigorization 1a. Cauchy on Limits and Continuity 1b. Cauchy on convergence 1c. Cauchy on the Radius of Convergence 2. Cauchy on the Derivative as a Limit 3. Cauchy on Maclaurin's Theorem 4. Cauchy-Moigno on the Fundamental Theorem of the Calculus B. Continuity and Integrability 5. Bolzano on Continuity and Limits 6. Riemann on Fourier Series and the Riemann Integral 7a. Heine Discusses Fourier Series 7b. Heine on the Foundations of Function Theory 8. Stieltjes on the Stieltjes Integral PART 2: FOUNDATIONS OF COMPLEX ANALYSIS A. Early Developments 9. Cauchy's Integral Theorem 10. Cauchy's Integral Formula 11. Cauchy's Calculus of Residues 12a. Cauchy on Liouville's Theorem 12b. Jordan on Liouville's Theorem B. Riemann's Influence 13. Riemann on the Cauchy-Riemann Equations 14. Riemann on Riemann Surfaces 15. Schwarz on Conformal Mapping PART 3: CONVERGENT EXPANSIONS A. The Convergence of Power Series 16. Gauss on the Hypergeometric Series 17. Abel on the Binomial Series B. The Influence of Weierstrass 18. Weierstrass on Analytic Functions of several Variables 19. Picard on Picard's Theorem 20a. Weierstrass on Infinite Products 20b. Mittag-Leffier's Theorem PART 4: ASYMPTOTIC EXPANSIONS A. Analytic Number Theory 21. Riemann on the Riemann Zeta Function 22. Hadamard on the Distribution of Primes B. Asymptotic Series 23. Stirling's Formula 24. Laplace on Generating Functions 25. Abel on the Laplace Transform 26. Poincare on Asymptotic Series 27. Lereh on Lerch's Theorem PART 5: FOURIER SERIES AND INTEGRALS A. Fourier Series 28. Fourier on Heat Flow in a Slab 29a. Fourier on Expansions in Sine Series 29b. Fourier on Heat Flow in a Ring 30. Dirichlet on the Convergence of Fourier Series 31. Wilbraham on the Gibbs Phenomenon 32. Fejre on the Convergence of Fourier Series B. The Fourier Integral 33a-b. Cauchy on the Fourier Integral 34. Fourier on the Fourier Integral 35. Cauchy on Linear Partial Differential Equations with Constant Coefficients PART 6: ELLIPTIC AND ABELIAN INTEGRALS 36. Legendre on Elliptic Integrals 37. Abel's Addition Theorem 38. Abel on Hyperelliptic Integrals 39a. Riemann on Abelian Integrals 39b. Roch on the Riemann-Roch Theorem PART 7: ELLIPTIC AND AUTOMORPHIC FUNCTIONS A. Elliptic and Hyperelliptic Functions 40. Abel on Elliptic Functions 41. Jacobi on Elliptic Functions 42. Jacobi on Some Identities 43. Jacobi on the Jacobi Theta Functions 44. Weierstrass's Al Functions B. Automorphic Functions 45. Poincare on Automorphic Functions 46. Klein on Fundamental Regions of Discontinuous Groups PART 8: ORDINARY DIFFERENTIAL EQUATIONS. I. A. Existence and Uniqueness Theorems 47. Cauchy on the Cauchy Polygon Method 48. Lipschitz on the Lipschitz Condition 49. Picard on the Picard Method 50. Osgood's Existence Theorem B. Sturm-Liouville Theory 51. Storm on Sturm's Theorems 52. Liouville on Sturm-Liouville Expansions. I. 53. Liouville on Sturm-Liouville Expansions. II. PART 9: ORDINARY DIFFERENTIAL EQUATIONS. II. A. Regular Singular Points 54. Fuchs on Isolated Singular Points 55. Frobenius on Regular Singular Points B. Other Fundamental Contributions 56. Lie on Groups of Transformations 57. Poincare on the Qualitative Theory of Differential Equations 58. Peano on the Peano Series PART 10: PARTIAL DIFFERENTIAL EQUATIONS A. The Cauchy-Kowalewski Theorem 59. Cauchy on the Cauchy-Kowalewski Theorem 60. Kowalewski on the Cauchy-Kowalewski Theorem B. Beginnings of Potential Theory 61. Laplace on the Laplacian Operator 62. Legendre on Legendre Polynomials 63. Poisson on the Poisson Equation C. Potential Theory Develops 64. Green on Green's Identities 65. Gauss on Potential Theory 66. Kelvin on Inversion PART 11: CALCULUS OF VARIATIONS A. Variational Principles of Dynamics 67. Lagrange on Properties Related to Least Action 68. Hamilton on Hamilton's Principle 69. Jacobi on the Hamilton-Jacobi Equations B. Intuitive Uses of Variational Principles 70a. Kelvin on the Dirichlet Principle 70b. Kelvin on a Variational Principle of Hydrodynamics 71a. Dirichlet on the Dirichiet Principle 71b. Rayleigh on the Rayleigh-Ritz Method C. Rigorous Existence Theorems 72. Du Bois-Reymond on the Fundamental Theorem of the Calculus of Variations 73. Poincare on His Methode tie Balayage 74. Hilbert on Dirichlet's Principle PART 12: WAVE EQUATIONS AND CHARACTERISTICS 75. Riemann on Plane Waves of Finite Amplitude 76. Helmholtz on the Helmholtz Equation 77. Kirchhoff's Identities for the Wave Equation 78. Volterra on Characteristics PART 13: INTEGRAL EQUATIONS 79. Abel's Integral Equation 80. Volterra on Inverting Integral Equations 81. Fredholm on the Theory of Integral Equations Short Bibliography Index
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