Book:K.G. Binmore/Mathematical Analysis: A Straightforward Approach

Subject Matter

 * Real Analysis

Contents

 * Preface


 * Real Numbers
 * Set notation
 * The set of real numbers
 * Arithmetic
 * Inequalities
 * Roots
 * Quadratic equations
 * Irrational numbers
 * Modulus


 * Continuum Property
 * Achilles and the tortoise
 * The continuum property
 * Supremum and infimum
 * Maximum and minimum
 * Intervals
 * Manipulations with $$\sup$$ and $$\inf$$


 * Natural Numbers
 * Introduction
 * Archimedean property
 * Principle of Induction


 * Convergent Series
 * The bulldozers and the bee
 * Sequences
 * Definition of convergence
 * Criteria for convergence
 * Monotone sequences
 * Some simple properties of convergent sequences
 * Divergent sequences


 * Subsequences
 * Subsequences
 * Bolzano-Weierstrass theorem
 * Lim sup and lim inf
 * Cauchy sequences


 * Series
 * Definitions
 * Series of Positive Terms
 * Elementary properties of series
 * Series and Cauchy sequences
 * Absolute and conditional convergence
 * Manipulations with series


 * Functions
 * Notation
 * Polynomial and rational functions
 * Combining functions
 * Inverse functions
 * Bounded functions


 * Limits of functions
 * Limits from the left
 * Limits from the right
 * $$f \left({x}\right) \to l$$ as $$x \to \xi$$
 * Continuity at a point
 * Connexion with convergent sequences
 * Properties of limits
 * Divergence


 * Continuity
 * Continuity on an interval
 * Continuity property


 * Differentiation
 * Derivatives
 * Higher derivatives
 * More notation
 * Properties of differentiable functions
 * Composite functions


 * Mean value theorems
 * Local maxima and minima
 * Stationary points
 * Mean value theorem
 * Taylor's theorem


 * Monotone functions
 * Definitions
 * Limits of monotone functions
 * Differentiable monotone functions
 * Inverse functions
 * Roots
 * Convex functions


 * Integration
 * Area
 * The integral
 * Some properties of the integral
 * Differentiation and integration
 * Riemann integral
 * More properties of the integral
 * Improper integrals
 * Euler-Maclaurin summation formula


 * Exponentiation and logarithm
 * Logarithm
 * Exponential
 * Powers


 * Power series
 * Interval of convergence
 * Taylor series
 * Continuity and differentiation


 * Trigonometric functions
 * Introduction
 * Sine and cosine
 * Periodicity


 * The gamma function
 * Stirling's formula
 * The gamma function
 * Properties of the gamma function


 * Appendix
 * This contains the proofs of 'propositions' left unproved in the main body of the text.