The calculus of finite differences:
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| Format: | Book |
| Language: | English |
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New York
Chelsea Publ. Comp.
1981
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| Edition: | 2., unaltered ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 558 S. |
| ISBN: | 0828403082 |
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| 100 | 1 | |a Milne-Thomson, Louis M. |d 1891-1979 |e Verfasser |0 (DE-588)115443924 |4 aut | |
| 245 | 1 | 0 | |a The calculus of finite differences |c by L. M. Milne-Thomson |
| 250 | |a 2., unaltered ed. | ||
| 264 | 1 | |a New York |b Chelsea Publ. Comp. |c 1981 | |
| 300 | |a 558 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Difference equations | |
| 650 | 4 | |a Finite differences | |
| 650 | 4 | |a Interpolation | |
| 650 | 0 | 7 | |a Differenzenrechnung |0 (DE-588)4149800-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Finite-Differenzen-Methode |0 (DE-588)4194626-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differenzenrechnung |0 (DE-588)4149800-8 |D s |
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| _version_ | 1804135867158102016 |
|---|---|
| adam_text | CONTENTS
PAGE
Introduction xxi
Notations xxiii
CHAPTER I
DIVIDED DIFFERENCES
1 0. Definitions 1
1 1. Newton s interpolation formula with divided differences 2
115. Rolle s Theorem 4
1 2. Remainder term in Newton s formula 5
1 3. Divided differences are symmetric functions of the arguments 7
1 31. Divided differences of x 7
1 4. Lagrange s interpolation formula 8
1 5. Expression of divided differences by means of determinants 9
1 6. Divided differences expressed by definite integrals 10
1 7. Divided differences expressed by contour integrals 11
1 8. Divided differences with repeated arguments 12
1 9. Interpolation polynomials 14
Examples I 17
CHAPTER II
DIFFERENCE OPERATORS
2 0. Difference notation 20
2 01. Central difference notation 22
2 1. Difference quotients 23
2105. Partial difference quotients 24
2 11. Difference quotients of factorial expressions 25
2 12. Expansion of a polynomial in factorials 27
2 13. Successive difference quotients of a polynomial 28
214. Difference quotients of a1 29
2 2. Properties of the operator A 30
2 3. The operator y 31
b ix
x CONTENTS
PAGE
2 4. The operator E 31
2 41. Herechel s Theorem 32
2 42. 4 {El°)ax = ax t {aa) 32
2 43. 4 {E )axu(x)= ax4 (a*E Mx) 32
2 5. Relations between A, E , D 33
2 51. The analogue of Leibniz Theorem 34
2 52. Difference quotients of axvx 35
2 53. Difference quotients of zero 36
2 54. Difference quotients in terms of derivates 37
2 6. The summation operator p 1 37
2 61. A theorem on the value of p, ^1 «4 39
2 62. Relation between sums and functional values 39
2 63. Moments 40
2 64. Partial summation 41
2 7. Summation of finite series 42
2 71. Summation of factorial expressions of the form xC ) 42
2 72. Polynomials 43
2 73. Factorial expressions of the form xfrm) 44
2.74. A certain type of rational function 45
2 75. The form ax 4 {x), 4 (x) a polynomial 46
2 76. The form vx f (x), f (x) a polynomial 46
2 77. Unclassified forms 48
Examples II 49
CHAPTER III
INTERPOLATION
3 0. Divided differences for equidistant arguments 56
3 1. Newton s interpolation formula (forward differences) 57
3 11. Newton s interpolation formula (backward differences) 59
3 12. The remainder term 61
3 2. Interpolation formulae of Gauss 63
3 3. Stirling s interpolation formula 67
3 4. BessePs interpolation formula 68
3 41. Modified Bessel s formula 71
3 5. Everett s interpolation formula 72
3 6. Steffensen s interpolation formula 74
3 7. Interpolation without differences 75
3 81. Aitken s linear process of interpolation by iteration 76
3 82. Aitken s quadratic process 78
3 83. Neville s process of iteration 81
Examples III 84
CONTENTS xi
CHAPTER IV
NUMERICAL APPLICATIONS OF DIFFERENCES
PAGE
4 0. Differences when the interval is subdivided 87
4 1. Differences of a numerical table 88
4 2. Subtabulation 91
4 3. Inverse interpolation 95
4 4. Inverse interpolation by divided differences 96
4 5. Inverse interpolation by iterated linear interpolation 97
4 6. Inverse interpolation by successive approximation 99
4 7. Inverse interpolation by reversal of series . . . 100
Examples IV 101
CHAPTER V
RECIPROCAL DIFFERENCES
5 1. Definition of reciprocal differences 104
5 2. Thiele s interpolation formula 106
5 3. Matrix notation for continued fractions 108
5 4. Reciprocal differences expressed by determinants 110
5 5. Reciprocal differences of a quotient 112
5 6. Some properties of reciprocal differences 114
5 7. The remainder in Thiele s formula 116
5 8. Reciprocal derivates ; the confluent case 117
5 9. Thiele s Theorem 119
Examples V 122
CHAPTER VI
THE POLYNOMIALS OF BERNOULLI AND EULER
6 0. The j polynomials 124
601. The £ polynomials 126
6 1. Definition of Bernoulli s polynomials 126
611. Fundamental properties of Bernoulli s polynomials 127
6 2. Complementary argument theorem 128
6 3. Relation between polynomials of successive orders 129
6 4. Relation of Bernoulli s polynomials to factorials 129
6 401. The integral of the factorial 131
6 41. Expansion of x(n in powers of x 133
6 42. Expansion of x in factorials 133
6 43. Generating functions of Bernoulli s numbers 134
xii CONTENTS
PAGE
6 5. Bernoulli s polynomials of the first order ... 136
6 501. Sum of the vth powers of the first n integers 137
6 51. Bernoulli s numbers of the first order 137
6 511. Euler Maclaurin Theorem for polynomials 139
6 52. Multiplication Theorem 141
6 53. Bernoulli s polynomials in the interval (0, 1) 141
6 6. The r, polynomials 142
6 7. Definition of Euler s polynomials 143
6 71. Fundamental properties of Euler s polynomials 144
6 72. Complementary argument theorem 145
6 73. Euler s polynomials of successive orders ... 145
6 8. Euler s polynomials of the first order 146
6 81. Euler s numbers of the first order 147
6 82. Boole s Theorem for polynomials 149
Examples VI 150
CHAPTER VII
NUMERICAL DIFFERENTIATION AND INTEGRATION
7 0. The first order derivate 154
7 01. Derivates of higher order 155
7 02. Markoff s formula 157
7 03. Derivates from Stirling s formula 159
7 04. Derivates from Bessel s formula 161
7 05. Differences in terms of derivates 162
7 1. Numerical integration 162
7 101. Mean Value Theorem 163
7 11. Integration by Lagrange s interpolation formula 164
7 12. Equidistant arguments 165
7 13. Remainder term, n odd 166
7 14. Remainder term, n even 167
7 2. Cotes formulae 168
7 21. Trapezoidal rule 170
7 22. Simpson s rule 171
7 23. Formulae of G. F. Hardy and Weddle 171
7 3. Quadrature formulae of the open type 172
7 31. Method of Gauss 173
7 33. Method of Tschebyscheff 177
7 4. Quadrature formulae involving differences . . . 180
7 41. Laplace s formula 181
7 42. Laplace s formula applied to differential equations 183
7 43. Central difference formulae 184
CONTENTS xiii
PAGE
7 5. Euler Maclaurin formula 187
7 51. Application to finite summation 191
7 6. Gregory s formula 191
7 7. Summation formula of Lubbock 193
Examples VII 196
CHAPTER VIII
THE SUMMATION PROBLEM
8 0. Definition of the principal solution or sum ... 201
8 1. Properties of the sum 204
8 11. Sum of a polynomial 208
8 12. Repeated summation 208
8 15. Proof of the existence of the principal solution (real variable) 209
8 16. Bernoulli s polynomials Bv (x oi) 213
82. Differentiation of the sum 213
8 21. Asymptotic behaviour of the sum for large values of x 214
8 22. Asymptotic behaviour of the sum for small values of o 216
8 3. Fourier series for the sum 218
84. Complex variable. Notation. Residue Theorem 220
8 41. Application of Cauchy s residue theorem 222
8 5. Extension of the theory 226
8 53. The sum of the exponential function 231
8 6. Functions with only one singular point 232
8 7. An expression for F(x | co) 238
Examples VIII 238
CHAPTER IX
THE PSI FUNCTION AND THE GAMMA FUNCTION
90.. The function ^(x a) 241
9 01. Differentiation of the Psi function 241
9 03. Partial and repeated summation 243
9 1. Asymptotic behaviour for large values of x 244
9 11. Partial fraction development of ir {x u) 245
92. Multiplication theorem for the Psi function 246
9 22. Fourier series for ¥ (x) 247
9 3. Gauss integral for ¥ (x) 247
9 32. Poisson s integral 248
9 4. Complementary argument theorem for the Psi function 249
9 5. The Gamma function 249
xiv CONTENTS
PAGE
9 52. Schlomilch s infinite product for T (x + 1) 250
9 53. Infinite products expressed by means of T (x) 251
9 54. Complementary argument theorem for T (x) 251
9 55. The residues of T(x) 252
9 56. Determination of the constant c 252
9 6. Stirling s series for log V (x + h) 253
9 61. An important limit 254
9 66. Generalised Gamma function r (x | co) 255
9 67. Some definite integrals 256
9 68. Multiplication theorem of the Gamma function 257
9 7. Euler s integral for T (x) 257
9 72. Complementary Gamma function T1(x) 258
9 8. Hypergeometric series and function F (a, b ; c ; x) ¦ 260
9 82. Hypergeometric function when x = 1 261
9 84. The Beta function B(ce, y) 262
9 86. Definite integral for the hypergeometric function 264
9 88. Single loop integral for the Beta function 265
9 89. Double loop integral for the Beta function 266
Examples IX 267
CHAPTER X
FACTORIAL SERIES
10 0. Associated factorial series 272
10 02. Convergence of factorial series 273
10 04. Region of convergence 275
10 06. Region of absolute convergence 276
10 07. Abel s identities 276
10 08. The upper limit of a sequence 277
10 09. Abscissa of convergence. Landau s Theorem 279
10 091. Majorant inverse factorial series 283
10 1. Series of inverse factorials 284
10 11. Uniform convergence of inverse factorial series 284
10 13. The poles of 0 (x) 287
10 15. Theorem of unique development 288
10 2. Application of Laplace s integral; generating function 288
10 22. Order of singularity and the convergence abscissa 292
10 3. The transformation (x, x + m) 293
10 32. The transformation (x, xfta) 294
10 4. Addition and multiplication of inverse factorial series 295
10 42. Differentiation of inverse factorial series 297
10 43. An asymptotic formula 298
CONTENTS xv
PAGE
10 44. Integration of inverse factorial series 299
10 5. Finite difference and sum of factorial series 300
10 6. Newton s factorial series 302
10 61. Uniform convergence of Newton s series 302
10 63. Null series 304
10 64. Unique development 305
10 65. Expansion in Newton s series ; reduced series ... 306
10 67. Abscissa of convergence of Newton s series 309
10 7. Majorant properties 310
10 8. Euler s transformation of series 311
10 82. Generating function 312
10 83. Laplace s integral and Newton s series 314
10 85. Expansion of the Psi function in Newton s series ... 315
10 9. Application to the hypergeometric function 316
Examples X 317
CHAPTER XI
THE DIFFERENCE EQUATION OF THE FIRST ORDER
110. Genesis of difference equations 322
11 01. The linear difference equation of the first order 324
11 1. The homogeneous linear equation 324
11 2. Solution by means of the Gamma function. Rational coefficients 327
11 3. Complete linear equation of the first order ... 328
11 31. The case of constant coefficients 329
11 32. Application of ascending continued fractions 330
11 33. Incomplete Gamma functions 331
11 34. Application of Prym s functions 332
11 4. The exact difference equation of the first order 334
11 41. Multipliers 339
11 42. Multipliers independent of x 339
11 43. Multipliers independent of u 340
11 5. Independent variable absent. Haldane s method 341
11 51. Boole s iterative method 343
11 6. Solution by differencing. Clairaut s form ... 344
11 7. Equations homogeneous in u 346
11 8. Riccati s form 346
11 9. Miscellaneous forms ........ 347
Examples XI 348
xvi CONTENTS
CHAPTER XII
GENERAL PROPERTIES OF THE LINEAR DIFFERENCE
EQUATION
PAGE
12 0. The homogeneous linear difference equation 351
12 01. Existence of solutions 352
12 1. Fundamental system of solutions 353
12 11. Casorati s Theorem 354
12 12. Heymann s Theorem 357
12 14. Relations between two fundamental systems ... 359
12 16. A critbuon for linear independence 360
12 2. Symbolic highest common factor 361
12 22. Symbolic lowest common multiple 363
12 24. Reducible equations 366
12 3. Reduction of order when a solution is known 367
12 4. Functional derivates 369
12 5. Multiple solutions of a difference equation .... 370
12 6. Multipliers. Adjoint equation 372
12 7. The complete linear equation. Variation of parameters 374
12 72. Polynomial coefficients 377
12 8. Solution by means of continued fractions ... 378
Examples XII 381
CHAPTER XIII
THE LINEAR DIFFERENCE EQUATION WITH CONSTANT
COEFFICIENTS
130. Homogeneous equations 384
1302. Boole s symbolic method 387
131. Complete equation 388
13 2. Boole s operational method 392
13 21. Case I, / (*) = xm 393
13 22. Case II, f {x) = ax 396
13 23. Caselll, cj (x) = axR(x) 398
13 24. The general case 398
13 25. Broggi s method for the particular solution .... 401
13 26. Solution by undetermined coefficients 403
13 3. Particular solution by contour integrals 404
13 32. Laplace s integral 407
13 4. Equations reducible to equations with constant coefficients 408
CONTENTS xvii
PAGE
13 5. Milne Thomson s operational method 410
13 51. Operations on unity 411
13 52. Operations on a given function X 412
13 53. Application to linear equations with constant coefficients 413
13 54. Simultaneous equations 415
13 55. Applications of the method 415
13 6. Simultaneous equations 420
13 7. Sylvester s non linear equations 420
13 8. Partial difference equations with constant coefficients 423
13 81. Alternative method 425
13 82. Equations resolvable into first order equations 426
13 83. Laplace s method 427
Examples XIII 429
CHAPTER XIV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. OPERATIONAL METHODS
14 0. The operator p 434
1401. The operator rc 436
1402. Inverse operations with tz 437
14 03. The operators nv px 439
141. Theorem I 439
1411. Theorem II 440
1412. Theorem III 440
1413. Theorem IV 442
14 14. Theorem V 443
14 2. Formal solution in series .. ... 445
14 21. Solution in Newton s series 448
14 22. Exceptional cases 451
14 3. Asymptotic forms of the solutions 457
14 31. Solutions convergent in a half plane on the left 459
14 4. The complete equation 460
14 5. Monomial difference equations ...... 4gj
14 6. Binomial equations ........ 455
14 7. Transformation of equations. Theorems VI, VII, VIII 467
14 71. Equation with linear coefficients 459
14 73. The equation (ax2+bx+c)u(x) + (ex+f)u(x l) +gu(x 2) = 0 472
14 75. The equation (ax2 + bx +c)Au + (ex +/)Am +gu = 0 474
14 8. Bronwin s method 475
14 9. Linear partial difference equations 475
xviii CONTENTS
PAGE
14 91. Laplace s method for partial equations 476
Examples XIV 477
CHAPTER XV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. LAPLACE S TRANSFORMATION
15 0. Laplace s transformation 478
151. Canonical systems of solutions 482
15 2. Factorial series for the canonical solutions ... 485
15 3. Asymptotic properties 487
15 31. Casorati s determinant 488
15 4. Partial fraction series 490
15 5. Laplace s difference equation 491
15 51. Reducible cases 493
15 52. Hypergeometric solutions 494
15 53. Partial fraction series 495
15 54. Relations between the canonical systems 496
15 55. The case a1=a2 498
15 6. Equations not of normal form 500
Examples XV 501
CHAPTER XVI
EQUATIONS WHOSE COEFFICIENTS ARE EXPRESSIBLE
BY FACTORIAL SERIES
160. Theorem IX 504
16 01. Theorem X 505
16 1. First normal form 508
16 2. Operational solution of an equation of the first normal form 509
16 3. Convergence of the formal solution 511
16 4. Example of solution 516
16 5. Second normal form 518
16 6. Note on the normal forms 519
Examples XVI 521
CHAPTER XVII
THE THEOREMS OF POINCARE AND PERRON
170. The linear equation with constant coefficients 523
171. Poincare s Theorem 526
CONTENTS xix
PAGE
17 2. Continued fraction solution of the second order equation 532
17 3. Sum equations 534
17 4. Homogeneous sum equations with constant coefiicients.
Theorem I 537
17 5. A second transformation of sum equations 539
17 6. General solution of sum equations. Theorem II 542
17 7. Difference equations of Poincare a type. Perron s Theorem 548
Examples XVII 550
Index 553
|
| adam_txt |
CONTENTS
PAGE
Introduction xxi
Notations xxiii
CHAPTER I
DIVIDED DIFFERENCES
1 0. Definitions 1
1 1. Newton's interpolation formula with divided differences 2
115. Rolle's Theorem 4
1 2. Remainder term in Newton's formula 5
1 3. Divided differences are symmetric functions of the arguments 7
1 31. Divided differences of x" 7
1'4. Lagrange's interpolation formula 8
1 5. Expression of divided differences by means of determinants 9
1 6. Divided differences expressed by definite integrals 10
1 7. Divided differences expressed by contour integrals 11
1 8. Divided differences with repeated arguments 12
1 9. Interpolation polynomials 14
Examples I 17
CHAPTER II
DIFFERENCE OPERATORS
2 0. Difference notation 20
2 01. Central difference notation 22
2 1. Difference quotients 23
2105. Partial difference quotients 24
2 11. Difference quotients of factorial expressions 25
2 12. Expansion of a polynomial in factorials 27
2 13. Successive difference quotients of a polynomial 28
214. Difference quotients of a1 29
2 2. Properties of the operator A 30
2 3. The operator y 31
b ix
x CONTENTS
PAGE
2 4. The operator E" 31
2 41. Herechel's Theorem 32
2 42. 4 {El°)ax = ax t {aa) 32
2 43. 4 {E'")axu(x)= ax4 (a*E"Mx) 32
2 5. Relations between A, E", D 33
2 51. The analogue of Leibniz' Theorem 34
2 52. Difference quotients of axvx 35
2 53. Difference quotients of zero 36
2 54. Difference quotients in terms of derivates 37
2 6. The summation operator p 1 37
2 61. A theorem on the value of p,"^1 «4 39
2 62. Relation between sums and functional values 39
2 63. Moments 40
2 64. Partial summation 41
2 7. Summation of finite series 42
2 71. Summation of factorial expressions of the form xC") 42
2 72. Polynomials 43
2 73. Factorial expressions of the form xfrm) 44
2.74. A certain type of rational function 45
2 75. The form ax 4 {x), 4 (x) a polynomial 46
2 76. The form vx f (x), f (x) a polynomial 46
2 77. Unclassified forms 48
Examples II 49
CHAPTER III
INTERPOLATION
3 0. Divided differences for equidistant arguments 56
3 1. Newton's interpolation formula (forward differences) 57
3 11. Newton's interpolation formula (backward differences) 59
3 12. The remainder term 61
3 2. Interpolation formulae of Gauss 63
3'3. Stirling's interpolation formula 67
3'4. BessePs interpolation formula 68
3 41. Modified Bessel's formula 71
3 5. Everett's interpolation formula 72
3 6. Steffensen's interpolation formula 74
3 7. Interpolation without differences 75
3 81. Aitken's linear process of interpolation by iteration 76
3'82. Aitken's quadratic process 78
3'83. Neville's process of iteration 81
Examples III 84
CONTENTS xi
CHAPTER IV
NUMERICAL APPLICATIONS OF DIFFERENCES
PAGE
4 0. Differences when the interval is subdivided 87
4 1. Differences of a numerical table 88
4 2. Subtabulation 91
4 3. Inverse interpolation 95
4 4. Inverse interpolation by divided differences 96
4 5. Inverse interpolation by iterated linear interpolation 97
4 6. Inverse interpolation by successive approximation 99
4 7. Inverse interpolation by reversal of series . . . 100
Examples IV 101
CHAPTER V
RECIPROCAL DIFFERENCES
5 1. Definition of reciprocal differences 104
5 2. Thiele's interpolation formula 106
5 3. Matrix notation for continued fractions 108
5 4. Reciprocal differences expressed by determinants 110
5 5. Reciprocal differences of a quotient 112
5'6. Some properties of reciprocal differences 114
5 7. The remainder in Thiele's formula 116
5'8. Reciprocal derivates ; the confluent case 117
5 9. Thiele's Theorem 119
Examples V 122
CHAPTER VI
THE POLYNOMIALS OF BERNOULLI AND EULER
6 0. The j polynomials 124
601. The £ polynomials 126
6 1. Definition of Bernoulli's polynomials 126
611. Fundamental properties of Bernoulli's polynomials 127
6 2. Complementary argument theorem 128
6 3. Relation between polynomials of successive orders 129
6 4. Relation of Bernoulli's polynomials to factorials 129
6 401. The integral of the factorial 131
6 41. Expansion of x(n in powers of x 133
6 42. Expansion of x" in factorials 133
6 43. Generating functions of Bernoulli's numbers 134
xii CONTENTS
PAGE
6 5. Bernoulli's polynomials of the first order . 136
6 501. Sum of the vth powers of the first n integers 137
6 51. Bernoulli's numbers of the first order 137
6 511. Euler Maclaurin Theorem for polynomials 139
6 52. Multiplication Theorem 141
6 53. Bernoulli's polynomials in the interval (0, 1) 141
6 6. The r, polynomials 142
6 7. Definition of Euler's polynomials 143
6 71. Fundamental properties of Euler's polynomials 144
6 72. Complementary argument theorem 145
6 73. Euler's polynomials of successive orders . 145
6 8. Euler's polynomials of the first order 146
6 81. Euler's numbers of the first order 147
6 82. Boole's Theorem for polynomials 149
Examples VI 150
CHAPTER VII
NUMERICAL DIFFERENTIATION AND INTEGRATION
7 0. The first order derivate 154
7 01. Derivates of higher order 155
7 02. Markoff's formula 157
7 03. Derivates from Stirling's formula 159
7 04. Derivates from Bessel's formula 161
7 05. Differences in terms of derivates 162
7 1. Numerical integration 162
7 101. Mean Value Theorem 163
7 11. Integration by Lagrange's interpolation formula 164
7 12. Equidistant arguments 165
7 13. Remainder term, n odd 166
7 14. Remainder term, n even 167
7 2. Cotes' formulae 168
7 21. Trapezoidal rule 170
7 22. Simpson's rule 171
7 23. Formulae of G. F. Hardy and Weddle 171
7 3. Quadrature formulae of the open type 172
7 31. Method of Gauss 173
7 33. Method of Tschebyscheff 177
7 4. Quadrature formulae involving differences . . . 180
7 41. Laplace's formula 181
7 42. Laplace's formula applied to differential equations 183
7 43. Central difference formulae 184
CONTENTS xiii
PAGE
7 5. Euler Maclaurin formula 187
7 51. Application to finite summation 191
7 6. Gregory's formula 191
7 7. Summation formula of Lubbock 193
Examples VII 196
CHAPTER VIII
THE SUMMATION PROBLEM
8 0. Definition of the principal solution or sum . 201
8 1. Properties of the sum 204
8 11. Sum of a polynomial 208
8 12. Repeated summation 208
8 15. Proof of the existence of the principal solution (real variable) 209
8 16. Bernoulli's polynomials Bv (x \ oi) 213
82. Differentiation of the sum 213
8 21. Asymptotic behaviour of the sum for large values of x 214
8'22. Asymptotic behaviour of the sum for small values of o 216
8 3. Fourier series for the sum 218
84. Complex variable. Notation. Residue Theorem 220
8 41. Application of Cauchy's residue theorem 222
8 5. Extension of the theory 226
8 53. The sum of the exponential function 231
8 6. Functions with only one singular point 232
8 7. An expression for F(x | co) 238
Examples VIII 238
CHAPTER IX
THE PSI FUNCTION AND THE GAMMA FUNCTION
90. The function ^(x\a) 241
9 01. Differentiation of the Psi function 241
9 03. Partial and repeated summation 243
9'1. Asymptotic behaviour for large values of x 244
9 11. Partial fraction development of 'ir {x \ u) 245
92. Multiplication theorem for the Psi function 246
9 22. Fourier series for ¥ (x) 247
9 3. Gauss' integral for ¥ (x) 247
9 32. Poisson's integral 248
9 4. Complementary argument theorem for the Psi function 249
9 5. The Gamma function 249
xiv CONTENTS
PAGE
9 52. Schlomilch's infinite product for T (x + 1) 250
9 53. Infinite products expressed by means of T (x) 251
9 54. Complementary argument theorem for T (x) 251
9 55. The residues of T(x) 252
9 56. Determination of the constant c 252
9 6. Stirling's series for log V (x + h) 253
9 61. An important limit 254
9 66. Generalised Gamma function r (x | co) 255
9 67. Some definite integrals 256
9 68. Multiplication theorem of the Gamma function 257
9 7. Euler's integral for T (x) 257
9 72. Complementary Gamma function T1(x) 258
9 8. Hypergeometric series and function F (a, b ; c ; x) ¦ 260
9 82. Hypergeometric function when x = 1 261
9 84. The Beta function B(ce, y) 262
9 86. Definite integral for the hypergeometric function 264
9 88. Single loop integral for the Beta function 265
9 89. Double loop integral for the Beta function 266
Examples IX 267
CHAPTER X
FACTORIAL SERIES
10 0. Associated factorial series 272
10 02. Convergence of factorial series 273
10 04. Region of convergence 275
10 06. Region of absolute convergence 276
10 07. Abel's identities 276
10 08. The upper limit of a sequence 277
10 09. Abscissa of convergence. Landau's Theorem 279
10 091. Majorant inverse factorial series 283
10 1. Series of inverse factorials 284
10 11. Uniform convergence of inverse factorial series 284
10 13. The poles of 0 (x) 287
10 15. Theorem of unique development 288
10 2. Application of Laplace's integral; generating function 288
10 22. Order of singularity and the convergence abscissa 292
10 3. The transformation (x, x + m) 293
10 32. The transformation (x, xfta) 294
10 4. Addition and multiplication of inverse factorial series 295
10 42. Differentiation of inverse factorial series 297
10 43. An asymptotic formula 298
CONTENTS xv
PAGE
10 44. Integration of inverse factorial series 299
10 5. Finite difference and sum of factorial series 300
10 6. Newton's factorial series 302
10 61. Uniform convergence of Newton's series 302
10 63. Null series 304
10 64. Unique development 305
10 65. Expansion in Newton's series ; reduced series . 306
10 67. Abscissa of convergence of Newton's series 309
10 7. Majorant properties 310
10 8. Euler's transformation of series 311
10 82. Generating function 312
10 83. Laplace's integral and Newton's series 314
10 85. Expansion of the Psi function in Newton's series . 315
10 9. Application to the hypergeometric function 316
Examples X 317
CHAPTER XI
THE DIFFERENCE EQUATION OF THE FIRST ORDER
110. Genesis of difference equations 322
11 01. The linear difference equation of the first order 324
11 1. The homogeneous linear equation 324
11 2. Solution by means of the Gamma function. Rational coefficients 327
11 3. Complete linear equation of the first order . 328
11 31. The case of constant coefficients 329
11 32. Application of ascending continued fractions 330
11 33. Incomplete Gamma functions 331
11 34. Application of Prym's functions 332
11 4. The exact difference equation of the first order 334
11 41. Multipliers 339
11 42. Multipliers independent of x 339
11 43. Multipliers independent of u 340
11 5. Independent variable absent. Haldane's method 341
11 51. Boole's iterative method 343
11 6. Solution by differencing. Clairaut's form . 344
11 7. Equations homogeneous in u 346
11 8. Riccati's form 346
11 9. Miscellaneous forms . 347
Examples XI 348
xvi CONTENTS
CHAPTER XII
GENERAL PROPERTIES OF THE LINEAR DIFFERENCE
EQUATION
PAGE
12 0. The homogeneous linear difference equation 351
12 01. Existence of solutions 352
12 1. Fundamental system of solutions 353
12 11. Casorati's Theorem 354
12 12. Heymann's Theorem 357
12 14. Relations between two fundamental systems . 359
12 16. A critbuon for linear independence 360
12 2. Symbolic highest common factor 361
12 22. Symbolic lowest common multiple 363
12 24. Reducible equations 366
12 3. Reduction of order when a solution is known 367
12 4. Functional derivates 369
12 5. Multiple solutions of a difference equation . 370
12 6. Multipliers. Adjoint equation 372
12 7. The complete linear equation. Variation of parameters 374
12 72. Polynomial coefficients 377
12 8. Solution by means of continued fractions . 378
Examples XII 381
CHAPTER XIII
THE LINEAR DIFFERENCE EQUATION WITH CONSTANT
COEFFICIENTS
130. Homogeneous equations 384
1302. Boole's symbolic method 387
131. Complete equation 388
13 2. Boole's operational method 392
13 21. Case I, / (*) = xm 393
13 22. Case II, f {x) = ax 396
13 23. Caselll, cj (x) = axR(x) 398
13 24. The general case 398
13 25. Broggi's method for the particular solution . 401
13 26. Solution by undetermined coefficients 403
13 3. Particular solution by contour integrals 404
13 32. Laplace's integral 407
13 4. Equations reducible to equations with constant coefficients 408
CONTENTS xvii
PAGE
13 5. Milne Thomson's operational method 410
13 51. Operations on unity 411
13 52. Operations on a given function X 412
13 53. Application to linear equations with constant coefficients 413
13 54. Simultaneous equations 415
13 55. Applications of the method 415
13 6. Simultaneous equations 420
13 7. Sylvester's non linear equations 420
13 8. Partial difference equations with constant coefficients 423
13 81. Alternative method 425
13 82. Equations resolvable into first order equations 426
13 83. Laplace's method 427
Examples XIII 429
CHAPTER XIV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. OPERATIONAL METHODS
14 0. The operator p 434
1401. The operator rc 436
1402. Inverse operations with tz 437
14 03. The operators nv px 439
141. Theorem I 439
1411. Theorem II 440
1412. Theorem III 440
1413. Theorem IV 442
14 14. Theorem V 443
14 2. Formal solution in series . . 445
14 21. Solution in Newton's series 448
14 22. Exceptional cases 451
14 3. Asymptotic forms of the solutions 457
14 31. Solutions convergent in a half plane on the left 459
14 4. The complete equation 460
14 5. Monomial difference equations . 4gj
14 6. Binomial equations . 455
14 7. Transformation of equations. Theorems VI, VII, VIII 467
14 71. Equation with linear coefficients 459
14 73. The equation (ax2+bx+c)u(x) + (ex+f)u(x l) +gu(x 2) = 0 472
14 75. The equation (ax2 + bx +c)Au + (ex +/)Am +gu = 0 474
14 8. Bronwin's method 475
14 9. Linear partial difference equations 475
xviii CONTENTS
PAGE
14 91. Laplace's method for partial equations 476
Examples XIV 477
CHAPTER XV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. LAPLACE'S TRANSFORMATION
15 0. Laplace's transformation 478
151. Canonical systems of solutions 482
15 2. Factorial series for the canonical solutions . 485
15 3. Asymptotic properties 487
15 31. Casorati's determinant 488
15 4. Partial fraction series 490
15 5. Laplace's difference equation 491
15 51. Reducible cases 493
15 52. Hypergeometric solutions 494
15 53. Partial fraction series 495
15 54. Relations between the canonical systems 496
15 55. The case a1=a2 498
15 6. Equations not of normal form 500
Examples XV ' 501
CHAPTER XVI
EQUATIONS WHOSE COEFFICIENTS ARE EXPRESSIBLE
BY FACTORIAL SERIES
160. Theorem IX 504
16 01. Theorem X 505
16 1. First normal form 508
16 2. Operational solution of an equation of the first normal form 509
16 3. Convergence of the formal solution 511
16 4. Example of solution 516
16 5. Second normal form 518
16 6. Note on the normal forms 519
Examples XVI 521
CHAPTER XVII
THE THEOREMS OF POINCARE AND PERRON
170. The linear equation with constant coefficients 523
171. Poincare's Theorem 526
CONTENTS xix
PAGE
17 2. Continued fraction solution of the second order equation 532
17 3. Sum equations 534
17 4. Homogeneous sum equations with constant coefiicients.
Theorem I 537
17 5. A second transformation of sum equations 539
17 6. General solution of sum equations. Theorem II 542
17 7. Difference equations of Poincare'a type. Perron's Theorem 548
Examples XVII 550
Index 553 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Milne-Thomson, Louis M. 1891-1979 |
| author_GND | (DE-588)115443924 |
| author_facet | Milne-Thomson, Louis M. 1891-1979 |
| author_role | aut |
| author_sort | Milne-Thomson, Louis M. 1891-1979 |
| author_variant | l m m t lmm lmmt |
| building | Verbundindex |
| bvnumber | BV021915918 |
| callnumber-first | Q - Science |
| callnumber-label | QA281 |
| callnumber-raw | QA281.M5 1981 |
| callnumber-search | QA281.M5 1981 |
| callnumber-sort | QA 3281 M5 41981 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 150 |
| ctrlnum | (OCoLC)7635646 (DE-599)BVBBV021915918 |
| dewey-full | 515/.62 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.62 |
| dewey-search | 515/.62 |
| dewey-sort | 3515 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | 2., unaltered ed. |
| format | Book |
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| id | DE-604.BV021915918 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:05:29Z |
| indexdate | 2024-07-09T20:47:19Z |
| institution | BVB |
| isbn | 0828403082 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015131089 |
| oclc_num | 7635646 |
| open_access_boolean | |
| owner | DE-706 |
| owner_facet | DE-706 |
| physical | 558 S. |
| publishDate | 1981 |
| publishDateSearch | 1981 |
| publishDateSort | 1981 |
| publisher | Chelsea Publ. Comp. |
| record_format | marc |
| spelling | Milne-Thomson, Louis M. 1891-1979 Verfasser (DE-588)115443924 aut The calculus of finite differences by L. M. Milne-Thomson 2., unaltered ed. New York Chelsea Publ. Comp. 1981 558 S. txt rdacontent n rdamedia nc rdacarrier Difference equations Finite differences Interpolation Differenzenrechnung (DE-588)4149800-8 gnd rswk-swf Finite-Differenzen-Methode (DE-588)4194626-1 gnd rswk-swf Differenzenrechnung (DE-588)4149800-8 s DE-604 Finite-Differenzen-Methode (DE-588)4194626-1 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015131089&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 |
| spellingShingle | Milne-Thomson, Louis M. 1891-1979 The calculus of finite differences Difference equations Finite differences Interpolation Differenzenrechnung (DE-588)4149800-8 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd |
| subject_GND | (DE-588)4149800-8 (DE-588)4194626-1 |
| title | The calculus of finite differences |
| title_auth | The calculus of finite differences |
| title_exact_search | The calculus of finite differences |
| title_exact_search_txtP | The calculus of finite differences |
| title_full | The calculus of finite differences by L. M. Milne-Thomson |
| title_fullStr | The calculus of finite differences by L. M. Milne-Thomson |
| title_full_unstemmed | The calculus of finite differences by L. M. Milne-Thomson |
| title_short | The calculus of finite differences |
| title_sort | the calculus of finite differences |
| topic | Difference equations Finite differences Interpolation Differenzenrechnung (DE-588)4149800-8 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd |
| topic_facet | Difference equations Finite differences Interpolation Differenzenrechnung Finite-Differenzen-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015131089&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT milnethomsonlouism thecalculusoffinitedifferences |