Springer Book Archives
1 Function Theory According to Chebyshev.- 1.1 Introduction.- 1.2 Functions of Minimal Deviation from Zero.- 1.2.1 The Lectures of A. A. Markov.- 1.2.2 E.I. Zolotaryov’s Problems; V. A. Markov’s Inequality.- 1.2.3 The Chebyshev Problem of Geographic Map Construction.- 1.3 Continued Fractions.- 1.3.1 Special Systems of Orthogonal Polynomials.- 1.3.2 Parameter-Dependence of the Zeros of Polynomials Obtained by Converting Series into Continued Fractions.- 1.3.3 Research on Bounds for Integrals.- 1.4 Conclusion.- 2 Ordinary Differential Equations.- 2.1 Summary of the Development of Ordinary Differetial Equations in the Eighteenth-Century.- 2.2 The Problem of Existence and Uniqueness.- 2.2.1 The Work of Cauchy.- 2.2.2 Development of the Method of Majorants.- 2.2.3 The Cauchy-Lipschitz Method.- 2.2.4 The Method of Successive Approximations.- 2.3 Integration of Equations in Quadratures.- 2.3.1 Liouville and the Riccati Equation.- 2.3.2 New Classes of Integrable Equations.- 2.3.3 Sophus Lie and the Problem of Integrability of Differential Equations in Quadratures.- 2.3.4 Singular Solutions.- 2.4 Linear Differential Equations.- 2.4.1 The General Theory.- 2.4.2 Boundary-value Problems. Sturm-Liouville Theory.- 2.4.3 Solution of Equations in Series and Special Functions.- 2.5 The Analytic Theory of Differential Equations.- 2.5.1 Origins of the Cauchy Theory. The Work of Briot and Bouquet.- 2.5.2 Bernhard Riemann.- 2.5.3 Lazarus Fuchs.- 2.5.4 Henri Poincaré.- 2.5.5 Nonlinear Equations.- 2.5.6 The Research of the Russian Mathematicians.- 2.5.7 Paul Painlevé.- 2.6 The Qualitative Theory of Differential Equations.- 2.6.1 Poincaré’s Qualitative Theory.- 2.6.2 Lyapunov’s Theory of Stability.- 2.6.3 Later Development of the Qualitative Theory of Differential Equations.- 2.6.4 Conclusion.- 3 The Calculus of Variations.- 3.1 Introduction.- 3.2 Calculus of Variations in the First Half of the Nineteenth Century.- 3.2.1 The Theory of Extrema of Multiple Integrals.- 3.2.2 The Hamilton-Jacobi Theory.- 3.2.3 Sufficient Conditions for a Weak Extremum.- 3.3 Calculus of Variations in the Second Half of the Nineteenth Century.- 3.3.1 The Proofs of the Jacobi Criterion and its Clarification. The Problem of Distinguishing Weak and Strong Extrema.- 3.3.2 Weierstrass’ Calculus of Variations.- 3.3.3 The Theory of the Simplest Variational Problem in the Second Half of the Nineteenth Century.- 3.3.4 The Creation of Field Theory.- 3.3.5 The Isoperimetric Problem.- 3.3.6 The Problems of Lagrange, Mayer, and Bolza.- 3.4 Conclusion. On Some Trends in the Development of the Calculus of Variations at the Turn of the Twentieth Century.- 4 The Calculus of Finite Differences.- 4.1 Interpolation.- 4.1.1 Finite Interpolation.- 4.1.2 Laplace’s Interpolation Series.- 4.1.3 Abel’s Interpolating Series.- 4.1.4 An Estimate of the Remainder in the Lagrange Interpolation Formula.- 4.1.5 Analytic Methods in the Theory of Interpolation.- 4.2 The Euler-Maclaurin Summation Formula.- 4.2.1 The Problem of Summation.- 4.2.2 Semiconvergent Series. Legendre’s Research.- 4.2.3 Poisson’s Derivation of the Summation Formula with Remainder.- 4.2.4 Abel’s Derivation.- 4.2.5 Jacobi’s Derivation. Enveloping Conditions.- 4.2.6 The Summation Formula according to Ostrogradskii.- 4.3 Finite-Difference Equations.- 4.3.1 Statement of the Problem. Summary of the Development of the Theory in the Eighteenth Century.- 4.3.2 Laplace’s Method.- 4.3.3 Poincaré’s Research.- 4.4 Conclusion.- Abbreviations Used for Multivolume and Periodical Publications.- A. General Works.- B. Collected Works and Classical Editions.- Literature to Part 1.- Literature to Part 2.- Literature to Part 3.- Literature to Part 4.- Index of Names.
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