Mathematical Analysis : Functions, Limits, Series, Continued Fractions.

✍ Scribed by Lyusternik, L. A.; Yanpol'Skii, A. R.

• Publisher: Elsevier Science
• Year: 2014
• Tongue: English
• Leaves: 407
• Series: International series in pure and applied mathematics.
• Category: Library

✦ Synopsis

Mathematical Analysis: Functions, Limits, Series, Continued Fractions provides an introduction to the differential and integral calculus. This book presents the general problems of the theory of continuous functions of one and several variables, as well as the theory of limiting values for sequences of numbers and vectors.

Organized into six chapters, this book begins with an overview of real numbers, the arithmetic linear continuum, limiting values, and functions of one variable. This text then presents the theory of series and practical methods of summation. Other chapters consider the theory of numerical series and series of functions and other analogous processes, particularly infinite continued fractions. This book discusses as well the general problems of the reduction of functions to orthogonal series. The final chapter deals with constants and the most important systems of numbers, including Bernoulli and Euler numbers.

This book is a valuable resource for mathematicians, engineers, and research workers.

✦ Table of Contents

Front Cover
Mathematical Analysis
Copyright Page
Table of Contents
FOREWORD
CHAPTER I. THE ARITHMETICAL LINEAR CONTINUUM AND FUNCTIONS DEFINED THERE
1. Real numbers and their representation
1. Real numbers
2. The numerical straight line
3. p-adic systems
4. Sets of real numbers
5. Bounded sets, upper and lower bounds
6. The theory of irrational numbers
2. Functions. Sequences
1. Functions of one variable
2. Upper and lower bounds of a function
3. Even and odd functions
4. Inverse functions
5. Periodic functions
6. Functional equations
7. Numerical sequences. 13. Uniform convergence of functions14. Convergence in the mean
15. The symbols o(x) and O(x)
16. Monotonic functions
17· Convex functions
CHAPTER II. n-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE
Introduction
1. n-dimensional spaces
1. n-dimensional coordinate space
2. n-dimensional vector space
3. Scalar product
4. A linear system and its basis
5. Linear functions
6. Linear envelope
7. Orthogonal systems of vectors
8. Biorthogonal systems of vectors
9. The projection of a vector on to a manifold
2. Passage to the limit, continuous functions and operators. 1. Passage to the limit in n-dimensional space2. Series of vectors
3. Continuous functions of n variables
4. Periodic functions of n variables. Manifolds of constancy
5. Passage to the limit for linear envelopes
6. Operators from En into Em
7. Iterative sequences
8. The principle of contraction mappings
3. Convex bodies in n-dimensional space
1. Fundamental definitions
2. Convex functions
3. Convex bodies and the norm of a vector
4. Support hyperplanes
5. Support functions and conjugate spaces
6. Fundamental theorems on support hyperplanes.

✦ Subjects

MATHEMATICS Mathematical Analysis analysis