Lyusternik / Yanpol'Skii Mathematical Analysis

Functions, Limits, Series, Continued Fractions
1. Auflage 2014
ISBN: 978-1-4831-9436-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 69, 418 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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.

Autoren/Hrsg.

Lyusternik, L. A.

Yanpol'Skii, A. R.

Inhaltsverzeichnis

1;Front Cover
;1
2;Mathematical Analysis
;4
3;Copyright Page
;5
4;Table of Contents
;6
5;FOREWORD;14
6;CHAPTER I. THE ARITHMETICAL LINEAR CONTINUUM AND FUNCTIONS DEFINED THERE;16
6.1;§ 1. Real numbers and their representation;16
6.1.1;1. Real numbers;16
6.1.2;2. The numerical straight line;17
6.1.3;3. p-adic systems;17
6.1.4;4. Sets of real numbers;19
6.1.5;5. Bounded sets, upper and lower bounds;21
6.1.6;6. The theory of irrational numbers;22
6.2;§ 2. Functions. Sequences;25
6.2.1;1. Functions of one variable;25
6.2.2;2. Upper and lower bounds of a function;26
6.2.3;3. Even and odd functions;28
6.2.4;4. Inverse functions;28
6.2.5;5. Periodic functions;29
6.2.6;6. Functional equations;30
6.2.7;7. Numerical sequences;31
6.2.8;8· Upper and lower bounds of a sequence;32
6.2.9;9. Maximum term of a sequence;32
6.2.10;10. Monotonie sequences;33
6.2.11;11. Double sequences;34
6.3;§ 3. Passage to the limit;34
6.3.1;1. The limit point of a set;34
6.3.2;2. The limit point and limit of a sequence;35
6.3.3;3· Fundamental theorems concerning limits;37
6.3.4;4. Some propositions on limits;38
6.3.5;5. Upper and lower limits of a sequence;39
6.3.6;6. Uniformly distributed sequences;40
6.3.7;7. Recurrent sequences;41
6.3.8;8. The symbols o(an) and O(an);42
6.3.9;9. Limit of a function;43
6.3.10;10· Right and left continuity of a function;43
6.3.11;11. Continuous and discontinuous functions;44
6.3.12;12· Functional sequences;45
6.3.13;13. Uniform convergence of functions;46
6.3.14;14. Convergence in the mean;48
6.3.15;15. The symbols o(x) and O(x);48
6.3.16;16. Monotonic functions;49
6.3.17;17· Convex functions;50
7;CHAPTER II. n-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE;53
7.1;Introduction;53
7.2;§ 1. n-dimensional spaces;54
7.2.1;1. n-dimensional coordinate space;54
7.2.2;2. n-dimensional vector space;55
7.2.3;3. Scalar product;56
7.2.4;4. A linear system and its basis;57
7.2.5;5. Linear functions;60
7.2.6;6. Linear envelope;63
7.2.7;7. Orthogonal systems of vectors;64
7.2.8;8. Biorthogonal systems of vectors;66
7.2.9;9. The projection of a vector on to a manifold;66
7.3;§ 2. Passage to the limit, continuous functions and operators;68
7.3.1;1. Passage to the limit in n-dimensional space;68
7.3.2;2. Series of vectors;71
7.3.3;3. Continuous functions of n variables;72
7.3.4;4. Periodic functions of n variables. Manifolds of constancy;77
7.3.5;5. Passage to the limit for linear envelopes;79
7.3.6;6. Operators from En into Em;80
7.3.7;7. Iterative sequences;82
7.3.8;8. The principle of contraction mappings;85
7.4;§3. Convex bodies in n-dimensional space;87
7.4.1;1. Fundamental definitions;87
7.4.2;2. Convex functions;88
7.4.3;3. Convex bodies and the norm of a vector;90
7.4.4;4. Support hyperplanes;91
7.4.5;5. Support functions and conjugate spaces;92
7.4.6;6. Fundamental theorems on support hyperplanes;94
7.4.7;7. The connection between reciprocal convex bodies;95
7.4.8;8. The cone. The tangent cone;96
7.4.9;9. Helly's theorem;97
7.4.10;10. Linear operations on sets;98
8;CHAPTER .II. SERIES;100
8.1;Introduction;100
8.2;1. Basic concepts;101
8.3;2. Some convergence tests for series;103
8.4;§ 1. Numerical series;105
8.4.1;1. Alternating series and series of constant sign;105
8.4.2;2. Properties of convergent series. The associative property;106
8.4.3;3. General tests for the convergence of series of positive terms;106
8.4.4;4. Remainder term estimates corresponding to the various convergence tests;108
8.4.5;5. Special tests for the convergence of series of positive terms.Estimates of the remainder term;110
8.4.6;6. The convergence of alternating series;118
8.4.7;7. Infinite products and their convergence;120
8.4.8;8. Double series. Fundamental concepts and definitions;124
8.4.9;9. Some properties of double series;126
8.4.10;10. Some convergence tests for double series of positive terms.Estimates of remainder term;128
8.5;§ 2. Series of functions;132
8.5.1;1. Fundamental properties and convergence tests;132
8.5.2;2. Power series;135
8.5.3;3. Operations on power series. Taylor series. Integration and differen tiation of power series;138
8.5.4;4. Complex series;144
8.5.5;5. Trigonometric Fourier series;147
8.5.6;6. Asymptotic series;155
8.5.7;7. Some methods of generalized summation of divergent series;157
8.6;§ 3· Methods of calculating the sum of a series;161
8.6.1;1. Elementary methods of exact summation;161
8.6.2;2. Summation of series with the aid of functions of a complex variable;163
8.6.3;3. Summation of series with the aid of Laplace transforms;165
8.6.4;4· Integral estimations for finite sums and infinite series;168
8.6.5;5. Rummer's transformation;171
8.6.6;6. Improvement of the convergence of series corresponding to agiven convergence test;172
8.6.7;7. Abel's transformation;176
8.6.8;8. A, N. Krylov's method of improving the convergence of trigonometric series;178
8.6.9;9. A. S. Maliev's method of improving the convergence of trigonometric series;182
9;CHAPTER IV. ORTHOGONAL SERIES AND ORTHOGONAL SYSTEMS;185
9.1;Introduction;185
9.2;§ 1. Orthogonal systems;187
9.2.1;1. Orthogonal systems of functions defined at n points;187
9.2.2;2. Orthogonal systems in En(x1, x2 . . . , xn);187
9.2.3;3. The best mean square approximation;189
9.2.4;4. Orthogonal systems of trigonometric functions;189
9.3;§ 2. General properties of orthogonal and biorthogonal systems;191
9.3.1;1. Orthogonality. Scalar (inner) product;191
9.3.2;2. Orthogonal systems of Bessel functions, Haar functions, etc.;194
9.3.3;3. Linear independence. The process of orthogonalization;201
9.3.4;4. Fourier coefficients. Closed systems;204
9.3.5;5· Fourier series in a trigonometric system;206
9.3.6;6. Biorthogonal systems of functions;209
9.4;§ 3. Orthogonal systems of polynomials;212
9.4.1;1. Zeros of orthogonal polynomials;214
9.4.2;2. Recurrence relations for orthogonal polynomials;214
9.4.3;3· Power moments. The expression of orthogonal polynomials interms of power moments;215
9.4.4;4. The connection between orthogonal polynomials and continued fractions;216
9.4.5;5. The conversion of orthogonal expansions into a sequence of approximating fractions;219
9.4.6;6. Orthogonal polynomials and quadrature formulae of the Gaussian type;221
9.4.7;7, The closure of an orthogonal system of polynomials;222
9.4.8;8. Christoffel's formula· The convergence of Fourier series in orthogonal polynomials;222
9.5;§ 4. Classical systems of orthogonal polynomials;225
9.5.1;1. Pearson's differential equation;225
9.5.2;2. The differentia] equations for corresponding classes of orthogonal polynomials;227
9.5.3;3. The expression, by means of the weight, of a polynomial of thenth degree belonging to an orthogonal system of polynomials;227
9.5.4;4. The generating function of an orthogonal system of polynomials with Pearson's weight;228
9.5.5;5. Legendre polynomials;229
9.5.6;6. Jacobi polynomials;234
9.5.7;7. Chebyshev polynomials of the first kind;238
9.5.8;8. Chebyshev polynomials of the second kind;245
9.5.9;9. Laguerre polynomials;248
9.5.10;10. Hermite polynomials;251
9.5.11;11. Chebyshev polynomials, orthogonal on a finitesystem of points;252
10;CHAPTER V. CONTINUED FRACTIONS;256
10.1;Introduction;256
10.2;1· Notation for continued fractions. Basic definitions;256
10.3;2. A brief historical note;257
10.4;§ 1. Continued fractions and their fundamental properties;257
10.4.1;1. The evaluation of convergents. Convergents;257
10.4.2;2. Transformations of continued fractions;259
10.4.3;3. Contraction and extension of continued fractions;260
10.4.4;4. The transformation of a continued fraction resulting from atheorem of Stolz;262
10.4.5;5. The properties of regular continued fractions;267
10.4.6;6. Equivalent and corresponding continued fractions;270
10.4.7;7. The formation of corresponding fractions. Viskovatov's method;272
10.4.8;8. Appell's method;274
10.5;§ 2. Fundamental tests for the convergence of continued fractions;276
10.5.1;1. The convergence of continued fractions;276
10.5.2;2. A necessary and sufficient test for the convergence of acontinued fraction in which the partial quotients havepositive terms (Seidel's test);279
10.5.3;3. Tests sufficent for the convergence of continued fractionsin which the partial quotients have positive terms;279
10.5.4;4. First set of tests sufficient for convergence;280
10.5.5;5. Tests for the convergence of continued fractions periodicin the limit;283
10.6;§ 3. The expansion of certain functions as continued fractions;284
10.6.1;1. Lagrange's method;284
10.6.2;2. Fundamental differential equation;285
10.6.3;3. The expansion of a power function as a continued fraction;286
10.6.4;4. The expansion of a logarithmic function as a continued fraction;287
10.6.5;5. The expansion of an exponential function as a continued fraction;288
10.6.6;6. Expansion of the function 7=arc tan x as a continued fraction;288
10.6.7;7. Expansion of the function y=.xodt/(1+tk) as a continued fraction;289
10.6.8;8. Expansion of tan x and tanh x as continued fractions;291
10.6.9;9. Expansion of Prima's function as a continued fraction;291
10.6.10;10. Expansion of the incomplete gamma fonction as acontinued fraction;293
10.6.11;11. Thiele's formula;293
10.6.12;12· Fractional approximations for sin x and sinh x;294
10.6.13;13. Fractional approximations for cos x and cosh x;295
10.6.14;14. Fractional approximation for the error function;296
10.6.15;15. Conversion of Stirling's series into a continued fraction;297
10.6.16;16. Fractional approximation for the gamma function;297
10.6.17;17. Fractional approximation for the logarithm of the gamma function;298
10.6.18;18. Fractional approximation for the derivative of thelogarithm of the gamma function;299
10.6.19;19. Obreshkov's formala;300
10.7;§ 4. Matrix methods;302
10.7.1;1· Extraction of the square root by means of second-order matrices;302
10.7.2;2. Solution of quadratic equations with the aid of second-order matrices;304
10.7.3;3. The connection between matrix methods and the theory of continued fractions;306
10.7.4;4. Hie reduction of quadratic surds to non-periodic continued fractions by means of second-order matrices with variable elements;308
10.7.5;5. Extraction of the root of any rational power by means of matrices;309
10.7.6;6. Solution of cubic equations by means of matrices;312
10.7.7;7. Recurrent series. The Bernoulli-Eider method;313
10.7.8;8. Connection between the Bernoulli–Eider method and matrix methods;315
10.7.9;9. Solution of higher degree equations by means of matrices;316
10.7.10;10. The idea behind Jacobfs algorithm;317
11;CHAPTER VI. SOME SPECIAL CONSTANTS AND FUNCTIONS;319
11.1;§ 1. Various constants and expressions;320
11.1.1;1. Some well-known constants;320
11.1.2;2. Soumne merical expressions;330
11.2;§ 2. Bernoulli and Euler numbers and polynomials;337
11.2.1;1. Bernoulli numbers and polynomials;337
11.2.2;2. Euler numbers and polynomials;347
11.3;§ 3. Elementary piecewise linear functions and delta-shaped functions;352
11.3.1;1. Piecewise linear functions;352
11.3.2;2. The ô (delta)-function;358
11.4;§ 4. Elementary special functions;361
11.4.1;1. Elliptic integrals;361
11.4.2;2. Integral functions;366
11.4.3;3. The error function;372
11.4.4;4. Fresnel integrals;374
11.4.5;5. Gamma and beta functions of Euler;377
11.4.6;6. Bessel functions;392
12;NOMENCLATURE;401
13;REFERENCES;405
14;INDEX;412
15;OTHER TITLES IN THE SERIES IN PUREAND APPLIED MATHEMATICS;418