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Author Courant, Richard. author Introduction to Calculus and Analysis [electronic resource] : Volume I / by Richard Courant, Fritz John New York, NY : Springer New York, 1989 http://dx.doi.org/10.1007/978-1-4613-8955-2 XXIII, 661 p. online resource

SUMMARY

From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text. In an additional pamphlet more problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given. This first volume of concerned primarily with functions of a single variable, whereas the second volume will discuss the more ramified theories of calculus (...)

CONTENT

1 Introduction -- 1.1 The Continuum of Numbers -- 1.2 The Concept of Function -- 1.3 The Elementary Functions -- 1.4 Sequences -- 1.5 Mathematical Induction -- 1.6 The Limit of a Sequence -- 1.7 Further Discussion of the Concept of Limit -- 1.8 The Concept of Limit for Functions of a Continuous Variable -- Supplements -- Problems -- 2 The Fundamental Ideas of the Integral and Differential Calculus -- 2.1 The Integral -- 2.2 Elementary Examples of Integration -- 2.3 Fundamental Rules of Integration -- 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral) -- 2.5 Logarithm Defined by an Integral -- 2.6 Exponential Function and Powers -- 2.7 The Integral of an Arbitrary Power of x -- 2.8 The Derivative -- 2.9 The Integral, the Primitive Function, and theFundamental Theorems of the Calculus -- Supplement The Existence of the Definite Integral of a Continuous Function -- Problems -- 3 The Techniques of Calculus -- A Differentiation and Integration of the Elementary Functions -- B Techniques of Integration -- C Further Steps in the Theory of Integral Calculus -- 4 Applications in Physics and Geometry -- 4.1 Theory of Plane Curves -- 4.2 Examples -- 4.3 Vectors in Two Dimensions -- 4.4 Motion of a Particle under Given Forces -- 4.5 Free Fall of a Body Resisted by Air -- 4.6 The Simplest Type of Elastic Vibration -- 4.7 Motion on a Given Curve -- 4.8 Motion in a Gravitational Field -- 4.9 Work and Energy -- Problems -- 5 Taylorโ{128}{153}s Expansion -- 5.1 Introduction: Power Series -- 5.2 Expansion of the Logarithm and the Inverse Tangent -- 5.3 Taylorโ{128}{153}s Theorem -- 5.4 Expression and Estimates for the Remainder -- 5.5 Expansions of the Elementary Functions -- 5.6 Geometrical Applications -- Appendix I -- Appendix II Interpolation -- Problems -- 6 Numerical Methods -- 6.1 Computation of Integrals -- 6.2 Other Examples of Numerical Methods -- 6.3 Numerical Solution of Equations -- Problems -- 7 Infinite Sums and Products -- 7.1 The Concepts of Convergence and Divergence -- 7.2 Tests for Absolute Convergence and Divergence -- 7.3 Sequences of Functions -- 7.4 Uniform and Nonuniform Convergence -- 7.5 Power Series -- 7.6 Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples -- 7.7 Power Series with Complex Terms -- Problems -- 8 Trigonometrie Series -- 8.1 Periodic Functions -- 8.2 Superposition of Harmonic Vibrations -- 8.3 Complex Notation -- 8.4 Fourier Series -- 8.5 Examples of Fourier Series -- 8.6 Further Discussion of Convergence -- 8.7 Approximation by Trigonometric and RationalPolynomials -- Appendix I -- Appendix II -- Problems -- 9 Differential Equations jor the SimplestTypes of Vibration -- 9.1 Vibration Problems of Mechanics and Physics -- 9.2 Solution of the Homogeneous Equation. Free Oscillations -- 9.3 The Nonhomogeneous Equation. Forced Oscillations -- List of Biographical Dates

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