A Formal Background to Mathematics 2a [and] 2b: A Critical Approach to Elementary Analysis [2 volume set]

Edwards, R. E.

ISBN 10: 0387905138 / ISBN 13: 9780387905136

Editore: Springer-Verlag, New York, 1980

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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.

Codice articolo 79339

Dati bibliografici

• Titolo: A Formal Background to Mathematics 2a [and] 2b: A Critical Approach to Elementary Analysis [2 volume set]
• Editore: Springer-Verlag, New York
• Data di pubblicazione: 1980
• Lingua: Inglese
• ISBN 10: 0387905138
• ISBN 13: 9780387905136
• Rilegatura: Paperback
• Condizione: Near Fine
• Cataloghi dei venditori:
  • Mathematics

Descrizione articolo

Riassunto

Book by Edwards R E

Contenuti

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.7.4.- X.7.6 General remarks concerning differential equations.- X.8 Calculus in several variables.- XI: Integration.- XI.1 Integration and area.- XI.1.1 Concept of area.- XI.1.2 Middle-of-the-road treatment.- XI.1.3 Area as basic concept.- XI.1.4 Purely analytic approach.- XI.1.5 Teaching background.- XI.1.6 Concerning terminology; hidden hypotheses.- XI.2 Analytic definition and study of integration.- XI.2.1 Partitions.- XI.2.2 Approximative sums.- XI.2.3 Definition of integrable functions; first consequences.- XI.2.4 Criterion of integrability; further remarks.- XI.2.5 Linearity of the integral.- XI.2.6 Integrability of continuous functions.- XI.2.7 Integrability of monotone functions.- XI.2.8 Integrability over subintervals.- XI.2.9 Additivity of the integral.- XI.2.10 Simplest cases of “change of variable”.- XI.2.11 A worked problem.- XI.2.12 Concerning the concept of integral.- XI.3 Integrals and primitives.- Preliminaries.- XI.3.1 Derivative of an integral; existence of a primitive.- XI.3.2 Remarks.- XI.3.3 Integral of a derivative.- XI.3.4 Tables of integrals.- XI.3.5 General comments.- XI.4 Integration by parts.- XI.4.1 Theorem.- XI.4.2 Remarks.- XI.5 Integration by change of variable (or by substitution).- XI.5.1 Theorem.- XI.5.2 Remarks.- XI.5.3 Use of XI.5.1.- XI.6 Termwise integration of sequences of functions.- Preliminaries.- XI.6.1 Convergence theorem for integrals.- XI.6.2 Comments on XI.6.1.- XI.6.3 Corollaries of XI.6.1.- XI.6.4 Ad hoc treatments.- XI.6.5 Problem.- XI.7 Improper integrals.- Preliminaries.- XI.7.1 Two problems discussed.- XI.7.2 Basic definitions and properties of certain improper integrals.- XI.7.3 More about conditionally convergent improper integrals.- XI.7.4 Generalised concept of limit.- XI.7.5 Concerning formalities.- XI.8 First order linear differential equations.- XI.8.1 The solutions of (1).- XI.8.2 Behaviour of solutions near the origin.- XI.8.3 Concerning formalities.- XI.9 Integrals in several variables.- XII: Complex Numbers: Complex Exponential and Trigonometric Functions.- XII.1 Definition of complex numbers.- XII.1.1 Basic definitions and theorems.- XII.1.2 Real and imaginary parts, absolute values and complex conjugates; rectangles, discs and circles.- XII.1.3 Generalisations of earlier theorems.- XII.1.4 Problems.- XII.2 Groups, subgroups and homomorphisms.- Preliminaries.- XII.2.1 Definition of groups.- XII.2.2 Subgroups.- XII.2.3 Homomorphisms and isomorphisms.- XII.2.4 Problems.- XII.3 Homomorphisms ofRinto ?; complex exponentials.- XII.3.1 Theorem.- XII.3.2 Theorem.- XII.3.3 Lemma.- XII.3.4 The functions e?.- XII.3.5 Remark.- XII.3.6 Homomorphisms of R into ?.- XII.3.7 Problems.- XII.3.8 Coupled differential equations for cos and sin.- XII.4 The exponential function with domainC.- XII.4.1 Definition.- XII.4.2 Properties of exp.- XII.4.3 Problems.- XII.4.4 Remarks.- XII.5 The trigonometric functions cosine and sine.- Preliminaries.- XII.5.1 Definition and first properties of cos and sin.- XII.5.2 Periodicity of cos and sin: the number ?.- XII.5.3 The ranges of sin, cos and ei.- XII.5.4 Polar representation, arguments, logarithms and powers.- XII.5.5 Problems.- XII.5.6 The function tan and a partial inverse.- XII.5.7 The irrationality of ?.- XII.5.8 Problems.- XII.5.9 The length of ?.- XII.5.10 The fundamental theorem of algebra.- XII.5.11 Remarks.- XII.6 Further inverse trigonometric functions.- XII.6.1 A partial inverse of sin.- XII.6.2 Discussion of a problem.- XII.6.3 Remarks.- XII.7 The simple harmonic equation.- XII.8 Another differential equation.- XII.9 Matrices and complex numbers.- XII.9.1 Definition of 2 × 2 real matrices.- XI.9.2 Complex numbers and 2 × 2 real matrices.- XII.9.3 Properties of M2(R).- XII.10 A glance at Fourier series.- XII.10.1 Fourier representation of functions.- XII.10.2 Approximation by trigonometric polynomials.- XII.10.3 Orhtonormality relations.- XII.10.4 Fourier series; the Dirichlet kernel.- XII.10.5 Theorem.- XII.10.6 The Fejér kernel.- XII.10.7 Fejér’s theorem.- XII.10.8 Parseval’s formula.- XII.10.9 Uniqueness theorem.- XII.10.10 Concluding remarks.- XII.11 Linear differential equations with constant coefficients.- XII.11.1 Concerning polynomials over C.- XII.11.2 Linear spaces and linear operators.- XII.11.3 Differential operators P(D).- XII.11.4 The structure of ker P(D).- XII.11.5 Description of K?,m.- XII.11.6 Complete solution of the homogeneous equation.- XII.11.7 The non-homogeneous equation.- XII.11.8 The spaces V, D+; convolutions over R.- XII.11.9 Green’s kernels.- XII.11.10 Solution of the non-homogeneous equation.- XII.11.11 Extensions.- XII.11.12 Systems of first order linear differential equations.- XII.11.13 Equations throughout intervals.- XII.11.14 Summary of procedure.- XI.11.15 Concerning formalities.- XIII: Concerning Approximate Integration.- XIII.1 Quotes from syllabus notes.- XIII.1.1 Error of the mid-ordinate rule.- XIII.1.2 Simpson’s rule versus the mid-ordinate rule.- XIII.1.3 Error a decreasing function of the number of strips.- XIII.2 Notation and preliminaries.- XIII.2.1 Mid-ordinate and Simpson’s Rules.- XIII.2.2 The sets C(r).- XIII.2.3 Approximation for continuous functions.- XIII.3 Precise formulation of statements XIII.1.1 – XIII.1.3.- XIII.3.1 Modification of XIII.1.1.- XIII.3.2 Modification of XIII.1.2.- XIII.3.3 Modification of XIII.1.3.- XIII.3.4 Discussion.- XIII.4 Some corrected versions.- XIII.4.1 Theorem.- XIII.4.2 Theorem.- XIII.4.3 Theorem.- XIII.4.4 Discussion.- XIII.4.5 Exceptional sets.- XIII.4.6 Total accuracy.- XIII.5 Falsity of statements XIII.3.1 – XIII.3.3.- Preliminaries.- XIII.5.1 Falsity of XIII.3.1.- XIII.5.2 Falsity of XIII.3.2.- XIII.5.3 Falsity of XIII.3.3.- XIII.5.4 The crucial point in XIII.5.2.- XIII.5.5 Concerning the proofs in XIII.5.1 – XIII.5.3.- XIII.5.6 Alternative refutations.- XIII.6 The formulas applied to tabulated data.- XIV: Differential Coefficients.- XIV.1 The d-notation and differential coefficients.- Preliminaries.- XIV.1.1 Differentials.- XIV.1.2 Differential coefficients.- XIV.1.3 Coordinate functions.- XIV.1.4 A more general approach to differential coefficients.- XIV.1.5 New definition of differential coefficients.- XIV.1.6 Basic properties of differential coefficients.- XIV.1.7 Differentiability and existence of the differential coefficient.- XIV.1.8 The chain rule again.- XIV.1.9 Fucntions with constant differential coefficients.- XIV.1.10 Summary.- XIV.1.11 Concerning extensions.- XIV.2 The simple harmonic equation.- XIV.2.1 Formulation of the problem.- XIV.2.2 The text book solution.- XIV.2.3 Criticism of the text book solution.- XIV.2.4 Alternative solutions.- XIV.2.5 Remark.- XV: Lengths of Curves.- XV.1 Quotes and criticisms.- XV.2 Paths.- XV.2.1 Definition of paths.- XV.2.2 Comments on the definition.- XV.2.3 Discussion: meagre sets.- XV.3 Lengths of paths.- XV.3.1 Definition of length.- XV.3.2 Monotonicity of L(?, p).- XV.3.3 Proof of (1) in XV.2.2.- XV.3.4 Length as a limit.- XV.4 Path length as an integral.- XV.4.1 Basic integral formula.- XV.4.2 A generalisation of XV.4.1.- XV.4.3 Further extensions.- XV.5 Ratio of arc length to chord length.- XV.5.1 Arcs.- XV.5.2 Ration of arc length to chord length.- XV.5.3 Cases in which XV.5.2(1) is true.- XV.5.4 Examples for which XV.5.2(1) is false.- XV.6 Additivity of arc length.- XV.7 Equivalent paths; simple paths.- XV.7.1 Equivalent paths.- XV.7.2 Simple paths.- XV.7.3 Theorem.- XV.7.4 Lemma.- XV.7.5 Reformulation of XV.7.4.- XV.7.6 Remarks.- XV.7.7 Lengths of simple paths and distance preserving maps.- XV.7.8 The shortest distance between two points.- XV.7.9 Minimising property of simple paths.- XV.7.10 Remarks on the preceding proof.- XV.8 Circular arcs; application to complex exponential and trigonometric functions.- XV.8.1 ? as a simple closed path.- XV.8.2 The arc length function and its inverse E.- XV.8.3 Further properties of E.- XV.8.4 E is a homomorphism.- XV.8.5 E is differentiable.- XV.8.6 The functions cos and sin.- XV.8.7 The truth of XV.5.2(1) for circular arcs.- XV.9 Angles and arguments.- XV.9.1 Arguments of complex numbers.- XV.9.2 Angles and their measurement.- XV.10 General remarks about curves.- XV.10.1 Characterisation of tracks of paths.- XV.10.2 The concept of curve.