A source book in classical analysis

Title page......Page 1
Preface......Page 5
A. Cauchy's Partial Rigorization......Page 13
la. Cauchy on Limits and Continuity......Page 14
1b. Cauchy on convergence......Page 15
2. Cauchy on the Derivative as a Limit......Page 16
3. Cauchy on Maclaurin's Theorem......Page 18
4. Cauchy-Moigno on the Fundamental Theorem of the Calculus......Page 20
B. Continuity and Integrability......Page 23
5. Bolzano on Continuity and Limits......Page 27
6. Riemann on Fourier Series and the Riemann Integral......Page 28
7a. Heine Discusses Fourier Series......Page 35
7b. Heine on the Foundations of Function Theory......Page 37
8. Stieltjes on the Stieltjes Integral......Page 38
A. Early Developments......Page 43
9. Cauchy's Integral Theorem......Page 45
10. Cauchy's Integral Formula......Page 49
11. Cauchy's Calculus of Residues......Page 52
12a. Cauchy on Liouville's Theorem......Page 56
12b. Jordan on Liouville's Theorem......Page 57
B. Riemann's Influence......Page 58
13. Riemann on the Cauchy-Riemann Equations......Page 60
14. Riemann on Riemann Surfaces......Page 62
15. Schwarz on Conformal Mapping......Page 68
A. The Convergence of Power Series......Page 72
16. Gauss on the Hypergeometric Series......Page 74
17. Abel on the Binomial Series......Page 80
B. The Influence of Weierstrass......Page 83
18. Weierstrass on Analytic Functions of several Variables......Page 86
19. Picard on Picard's Theorem......Page 91
20a. Weierstrass on Infinite Products......Page 92
20b. Mittag-Lefller's Theorem......Page 100
A. Analytic Number Theory......Page 105
21. Riemann on the Riemann Zeta Function......Page 107
22. Hadamard on the Distribution of Primes......Page 110
B. Asymptotic Series......Page 116
23. Stirling's Formula......Page 117
24. Laplace on Generating Functions......Page 120
25. Abel on the Laplace Transform......Page 127
26. Poincaré on Asymptotic Series......Page 129
27. Lerch on Lerch's Theorem......Page 137
A. Fourier Series......Page 142
28. Fourier on Heat Flow in a Slab......Page 144
29a. Fourier on Expansions in Sine Series......Page 150
29b. Fourier on Heat Flow in a Ring......Page 153
30. Dirichlet on the Convergence of Fourier Series......Page 157
31. Wilbraham on the Gibbs Phenomenon......Page 159
32. Fejér on the Convergence of Fourier Series......Page 162
B. The Fourier Integral......Page 168
33a-b. Cauchy on the Fourier Integral......Page 169
34. Fourier on the Fourier Integral......Page 176
35. Cauchy on Linear Partial Differential Equations with Constant Coefficients......Page 182
6. ELLIPTIC AND ABELIAN INTEGRALS......Page 187
36. Legendre on Elliptic IntegraIs......Page 189
37. Abel's Addition Theorem......Page 200
38. Abel on Hyperelliptic IntegraIs......Page 202
39a. Riemann on Abelian IntegraIs......Page 208
39b. Roch on the Riemann-Roch Theorem......Page 213
A. Elliptic and Hyperelliptic Functions......Page 216
40. Abel on Elliptic Functions......Page 218
41. Jacobi on Elliptic Functions......Page 219
42. Jacobi on Some Identities......Page 224
43. Jacobi on the Jacobi Theta Functions......Page 230
44. Weierstrass's Al Functions......Page 236
B. Automorphic Functions......Page 241
45. Poincaré on Automorphic Functions......Page 242
A. Existence and Uniqueness Theorems......Page 254
47. Cauchy on the Cauchy Polygon Method......Page 255
48. Lipschitz on the Lipschitz Condition......Page 259
49. Picard on the Picard Method......Page 262
50. Osgood's Existence Theorem......Page 263
B. Sturm-Liouville Theory......Page 270
51. Sturm on Sturm's Theorems......Page 271
52. Liouville on Sturm-Liouville Expansions. I......Page 280
53. Liouville on Sturm-Liouville Expansions. II......Page 288
A. Regular Singular Points......Page 294
54. Fuchs on Isolated Singular Points......Page 295
55. Frobenius on Regular Singular Points......Page 305
B. Other Fundamental Contributions......Page 311
56. Lie on Groups of Transformations......Page 312
57. Poincaré on the Qualitative Theory of DifferentiaI Equations......Page 317
58. Peano on the Peano Series......Page 323
A. The Cauchy-Kowalewski Theorem......Page 330
59. Cauchy on the Cauchy-Kowalewski Theorem......Page 331
60. Kowalewski on the Cauchy-Kowalewski Theorem......Page 339
B. Beginnings of Potential Theory......Page 347
61. Laplace on the Laplacian Operator......Page 348
62. Legendre on Legendre Polynomials......Page 350
63. Poisson on the Poisson Equation......Page 354
C. Potential Theory Develops......Page 358
64. Green on Green's Identities......Page 359
65. Gauss on Potential Theory......Page 370
66. Kelvin on Inversion......Page 374
A. Variational Principles of Dynamics......Page 377
67. Lagrange on Properties Related to Least Action......Page 379
68. Hamilton on Hamilton's Principle......Page 382
69. Jacobi on the Hamilton-Jacobi Equations......Page 386
B. Intuitive Uses of Variational Principles......Page 391
70a. Kelvin on the Dirichlet Principle......Page 392
70b. Kelvin on a Variational Principle of Hydrodynamics......Page 395
71a. Dirichlet on the Dirichlet Principle......Page 397
71b. Rayleigh on the Rayleigh-Ritz Method......Page 398
C. Rigorous Existence Theorems......Page 408
72. Du Bois-Reymond on the Fundamental Theorem of the Calculus of Variations......Page 404
73. Poincaré on His Méthode de Balayage......Page 407
74. Hilbert on Dirichlet's Principle......Page 411
12. WAVE EQUATIONS AND CHARACTERISTICS......Page 415
75. Riemann on Plane Waves of Finite Amplitude......Page 418
76. Helmholtz on the Helmholtz Equation......Page 426
77. Kirchhoff's Identities for the Wave Equation......Page 429
78. Volterra on Characteristics......Page 433
13. INTEGRAL EQUATIONS......Page 447
79. Abel's Integral Equation......Page 449
80. Volterra on Inverting Integral Equations......Page 454
81. Fredholm on the Theory of Integral Equations......Page 461
SHORT BIBLIOGRAPHY......Page 478
INDEX......Page 480