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A Source Book in Classical Analysis

Publisher: 
Harvard University Press
Number of Pages: 
486
Price: 
30.50
ISBN: 
0674822455
Date Received: 
Thursday, February 9, 2006
Reviewable: 
Yes
Include In BLL Rating: 
Yes
Garrett Birkhoff, editor
Publication Date: 
1973
Format: 
Hardcover
Audience: 
Category: 
Sourcebook
BLL Committee
02/19/2009
BLL Rating: 

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

Publish Book: 
Modify Date: 
Thursday, February 19, 2009

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