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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.

Tabla de contenidos provista por Syndetics

• 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
• 13a Cauchy on Liouville's Theorem
• 13b 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 Singula

Notas de autor provistas por Syndetics

Garrett Birkhoff was George Putnam Professor of Pure and Applied Mathematics at Harvard University and the author of a number of well-known books on mathematics and its applications.