A Formal Background to Mathematics 2a: A Critical Approach To Elementary Analysis (Universitext) - Softcover
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VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.2 Infinite limits.- VII.3 Subsequences.- VII.4 The Monotone Convergence Principle again.- VII.5 Suprema and infima of sets of real numbers.- VII.6 Exponential and logarithmic functions.- VII.7 The General Principle of Convergence.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.2 Properties of continuous functions.- VIII.3 General exponential, logarithmic and power functions.- VIII.4 Limit of a function at a point.- VIII.5 Uniform continuity.- VIII.6 Convergence of sequences of functions.- VIII.7 Polynomial approximation.- VIII.8 Another approach to expa.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.2 Absolute and conditional convergence.- IX.3 Decimal expansions.- IX.4 Convergence of series of functions.- X: Differentiation.- and hidden hypotheses.- X.1 Derivatives.- X.2 Rules for differentiation.- X.3 The mean value theorem and its corollaries.- X.4 Primitives.- X.5 Higher order derivatives.- X.6 Extrema and derivatives.- X.7 A differential equation and the exponential function again.- X.8 Calculus in several variables.- XI: Integration.- XI.1 Integration and area.- XI.2 Analytic definition and study of integration.- XI.3 Integrals and primitives.- XI.4 Integration by parts.- XI.5 Integration by change of variable (or by substitution).- XI.6 Termwise integration of sequences of functions.- XI.7 Improper integrals.- XI.8 First order linear differential equations.- XI.9 Integrals in several variables.- XII: Complex Numbers: Complex Exponential and Trigonometric Functions.- XII.1 Definition of complex numbers.- XII.2 Groups, subgroups and homomorphisms.- XII.3 Homomorphisms ofRinto?; complex exponentials.- XII.4 The exponential function with domainC.- XII.5 The trigonometric functions cosine and sine.- XII.6 Further inverse trigonometric functions.- XII.7 The simple harmonic equation.- XII.8 Another differential equation.- XII.9 Matrices and complex numbers.- XII.10 A glance at Fourier series.- XII.11 Linear differential equations with constant coefficients.- XIII: Concerning Approximate Integration.- XIII.1 Quotes from syllabus notes.- XIII.2 Notation and preliminaries.- XIII.3 Precise formulation of statements XIII.1.1 - XIII.1.3.- XIII.4 Some corrected versions.- XIII.5 Falsity of statements XIII.3.1 - XIII.3.3.- XIII.6 The formulas applied to tabulated data.- XIV: Differential Coefficients.- XIV.1 The d-notation and differential coefficients.- XIV.2 The simple harmonic equation.- XV: Lengths of Curves.- XV.1 Quotes and criticisms.- XV.2 Paths.- XV.3 Lengths of paths.- XV.4 Path length as an integral.- XV.5 Ratio of arc length to chord length.- XV.6 Additivity of arc length.- XV.7 Equivalent paths; simple paths.- XV.8 Circular arcs; application to complex exponential and trigonometric functions.- XV.9 Angles and arguments.- XV.10 General remarks about curves.
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A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis (Universitext)
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A Formal Background to Mathematics 2a: A Critical Approach To Elementary Analysis (Universitext)
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A Formal Background to Mathematics 2a [and] 2b: A Critical Approach to Elementary Analysis [2 volume set]
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Springer-Verlag, New York, 1980
ISBN 10: 0387905138
ISBN 13: 9780387905136
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Paperback. Zustand: Near Fine. Two volume set. 23.5 x 15.5 cm. xlvi 606pp, 607- 1170pp. Parts 2a and 2b only. Yellow softcovers. Toning to spines. 2a contains chapters for Hidden hypotheses, infinite limits, subsequences, The monotone Convergence principle, exponential and logarithmic functions, General principle of Convergence, Continuity and limits of functions, Convergence of series, Differentiation, Integration, complex numbers, approximate integration, differential coefficients, lengths of curves. 2b contains chapters on Line integrals, Segmental and triangular paths, Convex sets, Standard subdivision of triangular paths, Cauchy's theorem, Cauchy's integral formula, Logarithmic functions, Complex analysis, notations, problems and solutions. Universitext. Bestandsnummer des Verkäufers 79339
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A Formal Background to Mathematics 2a
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This item is printed on demand - it takes 3-4 days longer - Neuware -VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.- VII.1.8 Theorem.- VII.1.9 Problems.- VII.1.10 Theorem.- VII.1.11 Theorem.- VII.1.12 Examples.- VII.1.13 More about converses.- VII.2 Infinite limits.- VII.2.1 The symbols - , - ; the extended real line.- VII.2.2 Definition of convergence to or to - .- VII.2.3 Theorem.- VII.2.4 Remarks.- VII.2.5 Example.- VII.2.6 Problems.- VII.3 Subsequences.- VII.3.1 Definition of subsequences.- VII.3.2 Theorem.- VII.3.3 Theorem.- VII.3.4 Examples.- VII.3.5 Lemma.- VII.3.6 Remark.- VII.4 The Monotone Convergence Principle again.- VII.4.1 The MCP.- VII.4.2 Example: the compound interest sequence.- VII.4.3 Preliminaries concering the number e.- VII.4.4 Problems.- VII.4.5 Theorem (Weierstrass-Bolzano).- VII.4.6 Kronecker¿s Theorem.- VII.5 Suprema and infima of sets of real numbers.- VII.5.1 Suprema.- VII.5.2 Infima.- VII.5.3 Example.- VII.5.4 Problems.- VII.5.5 Concerning formalities.- VII.5.6 Concerning notation and terminology.- VII.6 Exponential and logarithmic functions.- VII.6.1 Definition of exp.- VII.6.2 Theorem.- VII.6.3 Theorem.- VII.6.4 Remarks.- VII.6.5 Theorem.- VII.6.6 Theorem.- VII.6.7 An alternative approach.- VII.6.8 Concerning formalities.- VII.7 The General Principle of Convergence.- VII.7.1 Definition.- VII.7.2 The GCP.- VII.7.3 Discussion of convergence principles.- VII.7.4 Remarks concerning Cantor¿s construction of R.- VII.7.5 Concerning existential proofs.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.1.1 Definition of continuous functions.- VIII.1.2 Examples.- VIII.1.3 Theorem.- VIII.1.4 Problems.- VIII.2 Properties of continuous functions.- VIII.2.1 Theorem (Intermediate Value Theorem).- VIII.2.2 Comments on the preceding proof.- VIII.2.3 Corollary.- VIII.2.4 A geometrical illustration.- VIII.2.5 Theorem.- VIII.2.6 Problems.- VIII.2.7 Theorem.- VIII.2.8 Corollary.- VIII.2.9 Remark.- VIII.2.10 Problem.- VIII.2.11 Remark.- VIII.2.12 Problems.- VIII.3 General exponential, logarithmic and power functions.- VIII.3.1 Real powers of positive numbers.- VIII.3.2 The exponential and logarithmic functions with base a.- VIII.3.3 Power functions.- VIII.3.4 Problems.- VIII.4 Limit of a function at a point.- VIII.4.1 Preliminary definitions.- VIII.4.2 The full and punctured limits of a function at a point.- VIII.4.3 Theorem.- VIII.4.4 Some formalities and further discussion.- VIII.4.5 Theorem.- VIII.4.6 Limits of composite functions.- VIII.4.7 Other species of limits; one sided limits.- VIII.4.8 Problems.- VIII.5 Uniform continuity.- VIII.5.1 Preliminary discussion.- VIII.5.2 Definition.- VIII.5.3 Theorem.- VIII.5.4 Problems.- VIII.5.5 Remarks.- VIII.6 Convergence of sequences of functions.- VIII.6.1 Definition of pointwise convergence.- VIII.6.2 Examples.- VIII.6.3 Further discussion.- VIII.6.4 Definition of uniform convergence.- VIII.6.5 Theorem.- VIII.6.6 Examples.- VIII.6.7 Theorem.- VIII.6.8 Theorem.- VIII.6.9 Discussion of some formalities.- VIII.7 Polynomial approximation.- VIII.7.1 Preliminaries.- VIII.7.2 Theorem (Weierstrass).- VIII.7.3 Theorem (Bernstein).- VIII.7.4 Remarks.- VIII.8 Another approach to expa.- Preliminaries.- VIII.8.1 Existence of a solution.- VIII.8.2 Uniqueness of the solution.- VIII.8.3 Summary.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.1.1 Definitions.- IX.1.2 Example.- IX.1.3 Theorem.- IX.1.4 Theorem.- IX.1.5 Theorem.- IX.1.6 Theorem.- IX.1.7 Examples.- IX.2 Absolute and conditional convergence.- IX.2.1 Definition of absolute and conditional convergence.- IX.2.2 Theorem.- IX.2.3 Theorem (General Comparison Test).- IX.2.4 Problems.- IX.2.5 Theorem (d¿Alembert¿s Ratio Test).- IX.2.6 Theorem (Cauchy n-th Root Test).- IX.2.7 Theorem (Leibnitz¿ Test).- IX.2.8 Problem.- IX.2.9 Theorem.- IX.2.10 Problems.- IX.2.11 General remarks.- IX.3 Decimal expansions.- IX.3.1 Lemma.- IX.3.2 Lemma.- IX.3.3 Corollary.- IX.3.4 Example.- IX.3.5 Liouville numbers.- IX.4 Convergence of series of functions.- IX.4.1 Theorem.- IX.4.2 Problems.- IX.4.3 Theorem.- IX.4.4 Remark.- IX.4.5 Concluding remarks.- X: Differentiation.- and hidden hypotheses.- X.1 Derivatives.- X.1.1 Definition of derivative.- X.1.2 The derivative function.- X.1.3 Comments on the definition of derivative.- X.1.4 Equivalent formulations of X.1.1.- X.1.5 Differentiability and continuity.- X.1.6 Local nature of differentiability.- X.1.7 Derivative of jn when $$n in dot Nx$$.- X.1.8 Derivative of a constant function.- X.2 Rules for differentiation.- X.2.1 Theorem.- X.2.2 Theorem (The chain rule).- X.2.3 Theorem.- X.2.4 Derivative of jr when r is rational.- X.2.5 Derivatives of exponential, logarithmic and general power functions.- X.2.6 Implicit algebraic functions.- X.2.7 Cauchy¿s ¿singular function¿.- X.2.8 Continuous nowhere differentiable functions.- X.2.9 Concerning routine exercises.- X.3 The mean value theorem and its corollaries.- X.3.1 Mean value theorem.- X.3.2 Remarks.- X.3.3 Corollary.- X.3.4 Remarks.- X.3.5 Relations with monotonicity.- X.4 Primitives.- X.4.1 Difference of two primitives.- X.4.2 The existence problem for primitives.- X.4.3 Functions with no primitive.- X.4.4 Darboux continuity.- X.5 Higher order derivatives.- X.6 Extrema and derivatives.- X.6.1 Extremum points.- X.6.2 Local extrema.- X.6.3 Theorem.- X.6.4 Theorem.- X.6.5 Theorem.- X.6.6 Remarks.- X.6.7 Global extrema.- X.6.8 Global Extrema (continued).- X.6.9 The case of rational functions.- X.6.10 Some examples.- X.7 A differential equation and the exponential function again.- X.7.1 A conventional approach.- X.7.2 Remarks.- X.7.3 Preferred approach.- X.7.4 The exponential function refounded.- X.7.5 Proof of (10) in X.7. 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A Formal Background to Mathematics 2a
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Zustand: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.1.1 Definition of convergence to zero.- VII.1.2 Remarks.- VII.1.3 Definition of convergence in R.- VII.1.4 Remarks.- VII.1.5 Lemma.- VII.1.6 Theorem.- VII.1.7 Theorem.-. Bestandsnummer des Verkäufers 5911689
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A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis
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ISBN 10: 0387905138
ISBN 13: 9780387905136
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