Functions of a Real Variable Elementary Theory

Name: Functions of a Real Variable Elementary Theory
Availability: OutOfStock
ISBN: 9783540653400

ISBN-10: 3540653406

ISBN-13: 9783540653400

Edition: 2004

Authors: P. Spain, Nicolas Bourbaki

List price: $159.00

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Description:

This monograph, written for mathematicians and third year graduates onwards, covers derivatives, primitives and integrals, elementary functions, differential equations, local study of functions, generalized Taylor expansions - Euler-MacLaurin summation formula, and the gamma function.

Book details

List price: $159.00
Copyright year: 2004
Publisher: Springer Berlin / Heidelberg
Publication date: 9/18/2003
Binding: Hardcover
Pages: 338
Size: 6.10" wide x 9.25" long x 0.75" tall
Weight: 1.584
Language: English

Nicolas Bourbaki is the pseudonym for a group of mathematicians that included Henri Cartan, Claude Chevalley, Jean Dieudonne, and Andres Weil. Mostly French, they emphasized an axiomatic and abstract treatment on all aspects of modern mathematics in Elements de mathematique. The first volume of Elements appeared in 1939. Subsequently, a wide variety of topics have been covered, including works on set theory, algebra, general topology, functions of a real variable, topological vector spaces, and integration. One of the goals of the Bourbaki series is to make the logical structure of mathematical concepts as transparent and intelligible as possible. The books listed below are typical of…



To the Reader


Contents



Derivatives



First Derivative



Derivative of a vector function



Linearity of differentiation



Derivative of a product



Derivative of the inverse of a function



Derivative of a composite function



Derivative of an inverse function



Derivatives of real-valued functions



The Mean Value Theorem



Rolle's Theorem



The mean value theorem for real-valued functions



The mean value theorem for vector functions



Continuity of derivatives



Derivatives of Higher Order



Derivatives of order n



Taylor's formula



Convex Functions of a Real Variable



Definition of a convex function



Families of convex functions



Continuity and differentiability of convex functions



Criteria for convexity


Exercises on ï¿½1


Exercises on ï¿½2


Exercises on ï¿½3


Exercises on ï¿½4



Primitives and Integrals



Primitives and Integrals



Definition of primitives



Existence of primitives



Regulated functions



Integrals



Properties of integrals



Integral formula for the remainder in Taylor's formula; primitives of higher order



Integrals over Non-Compact Intervals



Definition of an integral over a non-compact interval



Integrals of positive functions over a non-compact interval



Absolutely convergent integrals



Derivatives and Integrals of Functions Depending on a Parameter



Integral of a limit of functions on a compact interval



Integral of a limit of functions on a non-compact interval



Normally convergent integrals



Derivative with respect to a parameter of an integral over a compact interval



Derivative with respect to a parameter of an integral over a non-compact interval



Change of order of integration


Exercises on ï¿½1


Exercises on ï¿½2


Exercises on ï¿½3



Elementary Functions



Derivatives of the Exponential and Circular Functions



Derivatives of the exponential functions; the number e



Derivative of log<sub>a</sub> x



Derivatives of the circular functions; the number ï¿½



Inverse circular functions



The complex exponential



Properties of the function e<sup>z</sup>



The complex logarithm



Primitives of rational functions



Complex circular functions; hyperbolic functions



Expansions of the Exponential and Circular Functions, and of the Functions Associated with them



Expansion of the real exponential



Expansions of the complex exponential, of cos x and sin x



The binomial expansion



Expansions of log(1 + x), of Arc tan x and of Arc sin x


Exercises on ï¿½1


Exercises on ï¿½2


Historical Note (Chapters I-II-III)


Bibliography



Differential Equations



Existence Theorems



The concept of a differential equation



Differential equations admitting solutions that are primitives of regulated functions



Existence of approximate solutions



Comparison of approximate solutions



Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations



Continuity of integrals as functions of a parameter



Dependence on initial conditions



Linear Differential Equations



Existence of integrals of a linear differential equation



Linearity of the integrals of a linear differential equation



Integrating the inhomogeneous linear equation



Fundamental systems of integrals of a linear system of scalar differential equations



Adjoint equation



Linear differential equations with constant coefficients



Linear equations of order n



Linear equations of order n with constant coefficients



Systems of linear equations with constant coefficients


Exercises on ï¿½1


Exercises on ï¿½2


Historical Note


Bibliography



Local Study of Functions



Comparison of Functions on a Filtered Set



Comparison relations: I. Weak relations



Comparison relations: II. Strong relations



Change of variable



Comparison relations between strictly positive functions



Notation



Asymptotic Expansions



Scales of comparison



Principal parts and asymptotic expansions



Sums and products of asymptotic expansions



Composition of asymptotic expansions



Asymptotic expansions with variable coefficients



Asymptotic Expansions of Functions of a Real Variable



Integration of comparison relations: I. Weak relations



Application: logarithmic criteria for convergence of integrals



Integration of comparison relations: II. Strong relations



Differentiation of comparison relations



Principal part of a primitive



Asymptotic expansion of a primitive



Application to Series with Positive Terms



Convergence criteria for series with positive terms



Asymptotic expansion of the partial sums of a series



Asymptotic expansion of the partial products of an infinite product



Application: convergence criteria of the second kind for series with positive terms


Appendix



Hardyfields



Extension of a Hardy field



Comparison of functions in a Hardy field



(H) Functions



Exponentials and iterated logarithms



Inverse function of an (H) function


Exercises on ï¿½1


Exercises on ï¿½3


Exercises on ï¿½4


Exercises on Appendix



Generalized Taylor Expansions. Euler-Maclaurin Summation Formula



Generalized Taylor Expansions



Composition operators on an algebra of polynomials



Appell polynomials attached to a composition operator



Generating series for the Appell polynomials



Bernoulli polynomials



Composition operators on functions of a real variable



Indicatrix of a composition operator



The Euler-Maclaurin summation formula



Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers



Eulerian expansion of cot z



Eulerian expansion of sin z



Application to the Bernoulli numbers



Bounds for the Remainder in the Euler-Maclaurin Summation Formula



Bounds for the remainderin the Euler-Maclaurin Summation Formula



Application to asymptotic expansions


Exercises on ï¿½1


Exercises on ï¿½2


Exercises on ï¿½3


Historical Note (Chapters V and VI)


Bibliography



The Gamma Function



The Gamma Function in the Real Domain



Definition of the Gamma function



Properties of the Gamma function



The Euler integrals



The Gamma Function in the Complex Domain



Extending the Gamma function to C



The complements' relation and the Legendre-Gauss multiplication formula



Stirling's expansion


Exercises on ï¿½1


Exercises on ï¿½2


Historical Note


Bibliography


Index of Notation


Index