Kopp P.E. Analysis

Butterworth-Heinemann, 1996. — 200 p. — (Modular mathematics series). — ISBN: 0-340-64596-2, 978-0-340-64596-3.Building on the basic concepts through a careful discussion of covalence, (while adhering resolutely to sequences where possible), the main part of the book concerns the central topics of continuity, differentiation and integration of real functions. Throughout, the historical context in which the subject was developed is highlighted and particular attention is paid to showing how precision allows us to refine our geometric intuition. The intention is to stimulate the reader to reflect on the underlying concepts and ideas.Series PrefaceIntroduction: Why We Study Analysis
What the computer cannot see
From counting to complex numbers
From infinitesimals to limitsConvergent Sequences and Series
Convergence and summation
Algebraic and order properties of limitsCompleteness and Convergence
Completeness and sequences
Completeness and series
Alternating series
Absolute and conditional convergence of seriesFunctions Defined by Power Series
Polynomials – and what Euler did with them!
Multiplying power series: Cauchy products
The radius of convergence of a power series
The elementary transcendental functionsFunctions and Limits
Historical interlude: curves, graphs and functions
The modern concept of function: ordered pairs, domain and range
Combining real functions
Limits of real functions – what Cauchy meant!Continuous Functions
Limits that fit
Limits that do not fit: types of discontinuity
General power functions
Continuity of power seriesContinuity on Intervals
From interval to interval
Applications: fixed points, roots and iteration
Reaching the maximum: the Boundedness Theorem
Uniform continuity – what Cauchy meant?Differentiable Real Functions
Tangents: prime and ultimate ratios
The derivative as a limitMean Values and Taylor Series
The Mean Value Theorem
Tests for extreme points
L’Hôpitars Rules and the calculation of limits
Differentiation of power series
Taylor’s Theorem and series expansionsThe Riemann Integral
Primitives and the ‘arbitrary constant’
Partitions and step functions: the Riemann integral
Criteria for integrability
Classes of integrable functions
Properties of the integralIntegration Techniques
The Fundamental Theorem of the Calculus
Integration by parts and change of variable
Improper integrals
Convergent integrals and convergent seriesWhat Next? Extensions and Developments
Generalizations of completeness
Approximation of functions
Integrals of real functions: yet more completenessAppendix. Program Listings
Solutions to Exercises
Index