Mathematical analysis: a special course
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Pergamon Press
1965
|
| Edition: | 1. ed. |
| Series: | International series of monographs on pure and applied mathematics
77 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 485 S. graph. Darst. |
Staff View
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| 245 | 1 | 0 | |a Mathematical analysis |b a special course |c G. Ye. Shilov |
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Record in the Search Index
| _version_ | 1804116362781523968 |
|---|---|
| adam_text | CONTENTS
Foreword xiii
Chapter J
THE ARITHMETICAL LINEAR CONTINUUM AND FUNCTIONS
DEFINED THERE
§ 1. Real numbers and their representation 1
1. Real numbers 1
2. The numerical straight line 2
3. /;-adic systems 2
4. Sets of real numbers 4
5. Bounded sets, upper and lower bounds 6
6. The theory of irrational numbers 7
§ 2. Functions. Sequences 10
1. Functions of one variable 10
2. Upper and lower bounds of a function 11
3. Even and odd functions 13
4. Inverse functions 13
5. Periodic functions 14
6. Functional equations 15
7. Numerical sequences 16
8. Upper and lower bounds of a sequence 17
9. Maximum term of a sequence 17
10. Monotonic sequences 18
11. Double sequences 19
§ 3. Passage to the limit 19
1. The limit point of a set 19
2. The limit point and limit of a sequence 20
3. Fundamental theorems concerning limits 22
4. Some propositions on limits 23
5. Upper and lower limits of a sequence 24
6. Uniformly distributed sequences 25
7. Recurrent sequences 26
8. The symbols o(a.) and O(a,) 27
V
vi CONTENTS
9. Limit of a function 28
10. Right and left continuity of a function 28
11. Continuous and discontinuous functions 29
12. Functional sequences 30
13. Uniform convergence of functions 31
14. Convergence in the mean 33
15. The symbols o(x) and O(x) 33
16. Monotonic functions 34
17. Convex functions 35
Chapter II
b-DIMENSIONAL SPACES AND FUNCTIONS DEFINED THERE
Introduction 38
§ 1. /i-dimensional spaces 39
1. ^-dimensional coordinate space 39
2. /(-dimensional vector space 40
3. Scalar product 41
4. A linear system and its basis 42
5. Linear functions 45
6. Linear envelope 48
7. Orthogonal systems of vectors 49
8. Biorthogonal systems of vectors 51
9. The projection of a vector on to a manifold 51
§ 2. Passage to the limit, continuous functions and operators 53
1. Passage to the limit in ^-dimensional space 53
2. Series of vectors 56
3. Continuous functions of n variables 57
4. Periodic functions of n variables. Manifolds of constancy 62
5. Passage to the limit for linear envelopes 64
6. Operators from Em into Em 65
7. Iterative sequences 67
8. The principle of contraction mappings 70
§ 3. Convex bodies in n-dimensional space 72
1. Fundamental definitions 72
2. Convex functions 73
3. Convex bodies and the norm of a vector 75
4. Support hyperplanes 76
5. Support functions and conjugate spaces 77
CONTENTS vii
6. Fundamental theorems on support hyperplanes 79
7. The connection between reciprocal convex bodies 80
8. The cone. The tangent cone 81
9. Helly s theorem 82
10. Linear operations on sets 83
Chapter III
SERIES
Introduction 85
1. Basic concepts 86
2. Some convergence tests for series 88
§ 1. Numerical series 90
1. Alternating series and series of constant sign 90
2. Properties of convergent series. The associative property 91
3. General tests for the convergence of series of positive terms 91
4. Remainder term estimates corresponding to the various convergence
tests 93
5. Special tests for the convergence of series of positive terms. Estimates
of the remainder term. 95
6. The convergence of alternating series 103
7. Infinite products and their convergence 105
8. Double series. Fundamental concepts and definitions 109
9. Some properties of double series 111
10. Some convergence tests for double series of positive terms.
Estimates of remainder term 113
§ 2. Series of functions 117
1. Fundamental properties and convergence tests 117
2. Power series 120
3. Operations on power series. Taylor series. Integration and dif¬
ferentiation of power series 123
4. Complex series 129
5. Trigonometric Fourier series 132
6. Asymptotic series 140
7. Some methods of generalized summation of divergent series 142
§ 3. Methods of calculating the sum of a series 146
1. Elementary methods of exact summation 146
2. Summation of series with the aid of functions of a complex variable 148
3. Summation of series with the aid of Laplace transforms 150
viii CONTENTS
4. Integral estimations for finite sums and infinite series 153
5. Kummer s transformation 156
6. Improvement of the convergence of series corresponding to a given
convergence test 157
7. Abel s transformation 161
8. A. N. Krylov s method of improving the convergence of trigono¬
metric series 163
9. A. S. Maliyev s method of improving the convergence of trigono¬
metric series 163
Chapter IV
ORTHOGONAL SERIES AND ORTHOGONAL SYSTEMS
Introduction 170
§ 1. Orthogonal systems 172
1. Orthogonal systems of functions defined at « points 172
2. Orthogonal systems xaE%(xv xv ..., xn) 172
3. The best mean square approximation 174
4. Orthogonal systems of trigonometric functions 174
§ 2. General properties of orthogonal and biorthogonal systems 176
1. Orthogonality. Scalar (inner) product 176
2. Orthogonal systems of Bessel functions, Haar functions, etc. 179
3. Linear independence. The process of orthogonalization 186
4. Fourier coefficients. Closed systems 189
5. Fourier series in a trigonometric system 191
6. Biorthogonal systems of functions 194
§ 3. Orthogonal systems of polynomials 197
1. Zeros of orthogonal polynomials 199
2. Recurrence relations for orthogonal polynomials 199
3. Power moments. The expression of orthogonal polynomials in terms
of power moments 200
4. The connection between orthogonal polynomials and continued frac¬
tions 201
5. The conversion of orthogonal expansions into a sequence of approxi¬
mating fractions 204
6. Orthogonal polynomials and quadrature formulae of the Gaussian
type 206
7. The closure of an orthogonal system of polynomials 207
8. Christoflfel s formula. The convergence of Fourier series in ortho¬
gonal polynomials 207
CONTENTS ix
§ 4. Classical systems of orthogonal polynomials 210
1. Pearson s differential equation 210
2. The differential equations for corresponding classes of orthogonal
polynomials 212
3. The expression, by means of the weight, of a polynomial of the nth
degree belonging to an orthogonal system of polynomials 212
4. The generating function of an orthogonal system of polynomials
with Pearson s weight 213
5. Legendre polynomials 214
6. Jacobi polynomials 219
7. Chebyshev polynomials of the first kind 223
8. Chebyshev polynomials of the second kind 230
9. Laguerre polynomials 233
10. Hermite polynomials 236
11. Chebyshev polynomials, orthogonal on a finite system of points 237
Chapter V
CONTINUED FRACTIONS
Introduction 241
1. Notation for continued fractions. Basic definitions 241
2. A brief historical note 242
§ 1. Continued fractions and their fundamental properties 242
1. The evaluation of convergents. Convergents 242
2. Transformations of continued fractions 244
3. Contraction and extension of continued fractions 245
4. The transformation of a continued fraction resulting from a theorem
of Stolz 247
5. The properties of regular continued fractions 252
6. Equivalent and corresponding continued fractions 255
7. The formation of corresponding fractions. Viskovatov s method 257
8. Appell s method 259
§ 2. Fundamental tests for the convergence of continued fractions 261
1. The convergence of continued fractions 261
2. A necessary and sufficient test for the convergence of a continued
fraction in which the partial quotients have positive terms (Seidel s
test) 264
3. Tests sufficient for the convergence of continued fractions in which
the partial quotients have positive terms 264
X CONTENTS
4. First set of tests sufficient for convergence 265
5. Tests for the convergence of continued fractions periodic in the
limit 268
§ 3. The expansion of certain functions as continued fractions 269
1. Lagrange s method 269
2. Fundamental differential equation 270
3. The expansion of a power function as a continued fraction 271
4. The expansion of a logarithmic function as a continued fraction 272
5. The expansion of an exponential function as a continued fraction 273
6. Expansion of the function y = arc tan a: as a continued fraction 273
f* At
7. Expansion of the function v = = as a continued fraction 274
Jo1+
8. Expansion of tan x and tanh x as continued fractions 276
9. Expansion of Prima s function as a continued fraction 276
10. Expansion of the incomplete gamma function as a continued
fraction 278
11. Thiele s formula 278
12. Fractional approximations for sin x and sinh x 279
13. Fractional approximations for cos x and cosh x 280
14. Fractional approximation for the error function 281
15. Conversion of Stirling s series into a continued fraction 282
16. Fractional approximation for the gamma function 282
17. Fractional approximation for the logarithm of the gamma function 283
18. Fractional approximation for the derivative of the logarithm
of the gamma function 284
19. Obreshkov s formula 285
§ 4. Matrix methods 287
1. Extraction of the square root by means of second-order matrices 287
2. Solution of quadratic equations with the aid of second-order matrices 289
3. The connection between matrix methods and the theory of continued
fractions 291
4. The reduction of quadratic surds to non-periodic continued fractions
by means of second-order matrices with variable elements 293
5. Extraction of the root of any rational power by means of matrices 294
6. Solution of cubic equations by means of matrices 297
7. Recurrent series. The Bemoulli-Euler method 298
8. Connection between the Bernoulli-Euler method and matrix methods 300
9. Solution of higher degree equations by means of matrices 301
10. The idea behind Jacobi s algorithm 302
CONTENTS xi
Chapter VI
SOME SPECIAL CONSTANTS AND FUNCTION
§ 1. Various constants and expressions 305
1. Some well-known constants 305
2. Some numerical expressions 315
§ 2. Bernoulli and Euler numbers and polynomials 322
1. Bernoulli numbers and polynomials 322
2. Euler numbers and polynomials 332
§ 3. Elementary piecewise linear functions and delta-shaped functions 337
1. Piecewise linear functions 337
2. The a(delta)-function 343
§ 4. Elementary special functions 346
1. Elliptic integrals 346
2. Integral functions 351
3. The error function 357
4. Fresnel integrals 359
5. Gamma and beta functions of Euler 362
6. Bessel functions 377
Nomenclature 386
References 390
Index 397
Volumes published in the series in pure and applied mathematics 403
|
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| author_variant | g e š ge geš |
| building | Verbundindex |
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| language | English |
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| spelling | Šilov, Georgij E. 1917-1975 Verfasser (DE-588)133801780 aut Matematičeskij analiz Mathematical analysis a special course G. Ye. Shilov 1. ed. Oxford [u.a.] Pergamon Press 1965 XII, 485 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series of monographs on pure and applied mathematics 77 Analysis (DE-588)4001865-9 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Lineare Algebra (DE-588)4035811-2 s 2\p DE-604 Funktionalanalysis (DE-588)4018916-8 s 3\p DE-604 International series of monographs on pure and applied mathematics 77 (DE-604)BV001888024 77 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001254035&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Šilov, Georgij E. 1917-1975 Mathematical analysis a special course International series of monographs on pure and applied mathematics Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4018916-8 (DE-588)4035811-2 (DE-588)4151278-9 |
| title | Mathematical analysis a special course |
| title_alt | Matematičeskij analiz |
| title_auth | Mathematical analysis a special course |
| title_exact_search | Mathematical analysis a special course |
| title_full | Mathematical analysis a special course G. Ye. Shilov |
| title_fullStr | Mathematical analysis a special course G. Ye. Shilov |
| title_full_unstemmed | Mathematical analysis a special course G. Ye. Shilov |
| title_short | Mathematical analysis |
| title_sort | mathematical analysis a special course |
| title_sub | a special course |
| topic | Analysis (DE-588)4001865-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Analysis Funktionalanalysis Lineare Algebra Einführung |
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