In this post, we will see the book Mathematical Analysis – Functions, Limits, Series, Continued Fractions edited by L. A. Lyusternik; A. R. Yanpol’skii.

# About the book

The present book, together with its companion volume devoted to the differential and integral calculus, contains the fundamental part of the material dealt with in the larger courses of mathematical analysis. Included in this volume are general problems of the theory of continuous functions of one and several variables (with the geometrical basis of this theory), the theory of limiting values for sequences of numbers and vectors, and also the theory of numerical series and series of functions and other analogous infinite processes, in particular, infinite continued fractions.

The book was translated from Russian by D. E. Brown and edited by E. Spence. The book was published in 1965.

Credits to original uploader.

You can get the book here.

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# Contents

## CHAPTER I THE ARITHMETICAL LINEAR CONTINUUM AND FUNCTIONS DEFINED THERE 1

(L. A. Lyusternik and Ye. K. Isakova)

§ 1. Real numbers and their representation 1

§ 2. Functions. Sequences 10

§ 3. Passages of the limit 19

## CHAPTER II N-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE

(L. A. Lyusternik)

Introduction 38

§ 1. n-Dimensional spaces 39

§ 2. Passage to the limit, continuous function and operators 53

§ 3. Convex bodies in n-dimensional space 72

## Chapter III SERIES 85

(G. S. Salekhov and V. L. Danilov)

Introduction 85

§ 1. Numerical Series 90

§ 2. Series of functions 117

§ 3. Methods of calculating the sum of a series 146

## Chapter IV ORTHOGONAL SERIES AND ORTHOGONAL SYSTEMS

(A. N. Ivanova and L. A. Lyusternik),

Introduction 170

§ 1. Orthogonal Systems 172

§ 2. General properties of orthogonal and biorthogonal systems 176

§ 3. Orthogonal Systems of polynomials 197

§ 4. Classical systems of orthogonal polynomials 210

## Chapter V CONTINUED FRACTIONS

(author A. N. Khovanskii)

Introduction 241

§ 1. Continued fractions and their fundamental properties 242

§ 2. Fundamental tests for convergence of continued fractions 261

§ 3. The expansion of certain functions as continued fractions 269

§ 4. Matrix Methods 287

## Chapter VI SOME SPECIAL CONSTANTS AND FUNCTIONS

(L. A. Lyusternik, L. Ya. Tslaf and A. R. Yanpol’skii)

§ 1. Various constants and expressions 305

§ 2. Bernoulli and Euler numbers and polynomials 322

§ 3. Elementary piecewise linear functions and delta-shaped functions 337

NOMENCLATURE 386

REFERENCES 390

INDEX 397