Table of Contents

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. Poincaré 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. Fejré 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. Poincaré 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. Poincaré 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. Poincaré 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