Calculus of finite differences:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Chelsea Publ. Co.
1965
|
| Ausgabe: | 3. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 654 S. graph. Darst. |
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| 100 | 1 | |a Jordan, Charles |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Calculus of finite differences |c by Charles Jordan |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York, NY |b Chelsea Publ. Co. |c 1965 | |
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Datensatz im Suchindex
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| adam_text | CONTENTS.
Chapter I. On Operations.
§ 1. Historical and Bibliographical Notes .... 1
§ 2. Definition of differences 2
§ 3. Operation of displacement 5
§ 4. Operation of the mean 6
§ 5. Symbolical Calculus 7
§ 6. Symbolical methods 8
§ 7. Receding Differences 14
§ 8. Central Differences 15
§ 9. Divided Differences 18
§ 10. Generating functions 20
§ 11. General rules to determine generating functions . 25
§ 12. Expansion of functions into power series ... 29
§ 13. Expansion of function by aid of decomposition into
partial fractions 34
§ 14. Expansion of functions by aid of complex integrals . 40
§ 15. Expansion of a function by aid of difference equations 41
Chapter II. Functions important in the Calculus of Finite
Differences.
§ 16. The Factorial 45
§ 17. The Gamma function 53
§ 18. Incomplete Gamma function 56
§ 19. The Digamma function 58
§ 20. The Trigamma function 60
§ 21. Expansion of logr(x M) into a power series . . 61
§ 22. The Binomial coefficient • 62
§ 23. Expansion of a function into a series of binomial
coefficients
xii
§ 24. Beta functions 80
§ 25. Incomplete Beta functions 83
§ 26. Exponential functions 87
§ 27. Trigonometric functions 88
§ 28. Alternate functions 92
§ 29. Functions whose differences or means are equal to
zero 94
§ 30. Product of two functions. Differences .... 94
§ 31. Product of two functions. Means 98
Chapter 111. Inverse Operation of Differences and Means. Sums.
§ 32. Indefinite sums 100
§ 33. Indefinite sum obtained by inversion .... 103
§ 34. Indefinite sum obtained by summation by parts . . 105
§ 35. Summation by parts of alternate functions . . .108
§ 36. Indefinite sums determined by difference equations . 109
§ 37. Differences, sums and means of infinite series . , 110
§ 38. Inverse operation of the mean Ill
§ 39. Other methods of obtaining inverse means . . ,113
§ 40. Sums 116
§ 41. Sums determined by indefinite sums 117
§ 42. Sum of reciprocal factorials by indefinite sums . . 121
§ 43. Sums of exponential and trigonometric functions . 123
§ 44. Sums of other Functions 129
§ 45. Determination of sums by symbolical formulae . . 131
§ 46. Determination of sums by generating functions . .136
§ 47. Determination of sums by geometrical considerations 138
§ 48. Determination of sums by the Calculus of Probability 140
§ 49. Determination of alternate sums starting from usual
sums 140
Chapter IV. Stirling s Numbers.
§ 50. Expansion of factorials into power series. Stirling s
numbers of the first kind 142
§ 51. Determination of the Stirling numbers starting from
their definition 145
§ 52. Resolution of the difference equations . . . .147
xiii
§ 53. Transformation of a multiple sum without repeti¬
tion into sums without restriction 153
§ 54. Stirling s numbers expressed by sums. Limits . . 159
§ 55. Application of the Stirling numbers of the first kind 163
§ 56. Derivatives expressed by differences . . . .164
§ 57. Stirling numbers of the first kind obtained by proba¬
bility 166
§ 58. Stirling numbers of the second kind 168
§ 59. Limits of expressions containing Stirling numbers of
the second kind 173
§ 60. Generating functions of the Stirling numbers of the
second kind 174
§ 61. Stirling numbers of the second kind obtained by
probability 177
§ 62. Decomposition of products of prime numbers into
factors 179
§ 63. Application of the expansion of powers into series
of factorials 181
§ 64. Formulae containing Stirling numbers of both kinds 182
§ 65. Inversion of sums and series. Sum equations . . 183
§ 66. Deduction of certain formulae containing Stirling
numbers 185
§ 67. Differences expressed by derivatives .... 189
§ 68. Expansion of a reciprocal factorial into a series of
reciprocal powers and vice versa 192
§ 69. The operation 0 195
§ 70. The operation W 199
§ 71. Operations AmD m and DmA m 200
§ 72. Expansion of a function of function by aid of Stirling
numbers. Semi invariants of Thiele 204
§ 73. Expansion of a function into reciprocal factorial
series and into reciprocal power series .... 212
§ 74. Expansion of the function 1/y into a series of
powers of x 216
§ 75. Changing the origin of the intervals 219
§ 76. Changing the length of the interval 220
§ 77. Stirling s polynomials 224
xiv
Chapter V. Bernoulli Polynomials and Numbers.
§ 78. Bernoulli polynomials of the first kind .... 230
§ 79. Particular cases of the Bernoulli polynomials . . 236
§ 80. Symmetry of the Bernoulli polynomials .... 238
§ 81. Operations performed on the Bernoulli polynomial . 240
§ 82. Expansion of the Bernoulli polynomial into a Fourier
series. Limits. Sum of reciprocal power series . . 242
§ 83. Application of the Bernoulli polynomials . . . 246
§ 84. Expansion of a polynomial into Bernoulli polynomials 248
§ 85. Expansion of functions into Bernoulli polynomials.
Generating functions 250
§ 86. Raabe s multiplication theorem of the Bernoulli
polynomials 252
§ 87. The Bernoulli series 253
§ 88. The Maclaurin Euler summation formula . . . 260
§ 89. Bernoulli polynomials of the second kind . . . 265
§ 90. Symmetry of the Bernoulli polynomials of the second
kind 268
§ 91. Extrema of the polynomials 269
§ 92. Particular cases of the polynomials .... 272
§ 93. Limits of the numbers bn and of the polynomials yn [x] 272
§ 94. Operations on the Bernoulli polynomials of the
second kind 275
§ 95. Expansion of the Bernoulli polynomials of the
second kind 276
§ 96. Application of the polynomials 277
§ 97. Expansion of a polynomial into a series of Bernoulli
polynomials of the second kind 277
§ 98. The Bernoulli series of the second kind .... 280
§ 99. Gregory s summation formula 284
Chapter VI. Euler s and Boole s polynomials. Sums of
reciprocal powers.
§ 100. Euler s polynomials 288
§ 101. Symmetry of the Euler polynomials .... 292
§ 102. Expansion of the Euler polynomials into a series
of Bernoulli polynomials of the first kind . . . 295
XV
§ 103. Operations on the Euler polynomials .... 296
§ 104. The Tangent coefficients 298
§ 105. Euler numbers 300
§ 106. Limits of the Euler polynomials and numbers . . 302
§ 107. Expansion of the Euler polynomials into Fourier
series 303
§ 108. Application of the Euler polynomials .... 306
§ 109. Expansion of a polynomial into a series of Euler
polynomials 307
§ 110. Multiplication theorem of the Euler polynomials . 311
§ 111. Expansion of a function into an Euler series . . 313
§ 112. Boole s first summation formula 315
§ 113. Boole s polynomials 317
§ 114. Operations on the Boole polynomials. Differences . 320
§ 115. Expansion of the Boole polynomials into a series of
Bernoulli polynomials of the second kind . . . 321
§ 116. Expansion of a function into Boole polynomials . 322
§ 117. Boole s second summation formula 323
§ 118. Sums of reciprocal powers. Sum of 1/x by aid of
the digamma function 325
§ 119. Sum of 1/x2 by aid of the trigamma function . . 330
§ 120. Sum of a rational fraction 335
§ 121. Sum of reciprocal powers. Sum of xm .... 338
§ 122. Sum of alternate reciprocal powers by the /M*)
function 347
Chapter VII. Expansion of Functions, Interpolation,
Construction of Tables.
§ 123. Expansion of a Function into a series of polynomials 355
§ 124. The Newton series 357
§ 125. Interpolation by aid of Newton s Formula and
Construction of Tables 360
§ 126. Inverse interpolation by Newton s formula . . 366
§ 127. Interpolation by the Gauss series 368
§ 128. The Bessel and the Stirling series 373
§ 129. Everett s formula 376
§ 130. Inverse interpolation by Everett s formula . . 381
§ 131. Lagrange s interpolation formula 385
xvi
§ 132. Interpolation formula without printed differences.
Remarks on the construction of tables .... 390
§ 133. Inverse Interpolation by aid of the formula of the
preceding paragraph 411
§ 134. Precision of the interpolation formulae . . . .417
§ 135. General Problem of Interpolation 420
Chapter VIII. Approximation and Graduation.
§ 136. Approximation according to the principle of
moments 422
§ 137. Examples of function chosen 426
§ 138. Expansion of a Function into a series of Legendre s
polynomials 434
§ 139. Orthogonal polynomials with respect to x=jc0,
*! **¦_! 426
§ 140. Mathematical properties of the orthogonal poly¬
nomials 442
§ 141. Expansion of a function into a series of orthogonal
polynomials 447
§ 142. Approximation of a function given for i—0,1, 2, . .. ,
N—l 451
§ 143. Graduation by the method of least squares . . 456
§ 144. Computation of the binomial moments .... 460
§ 145. Fourier series 463
§ 146. Approximation by trigonometric functions of dis¬
continuous variables 465
§ 147. Hermite polynomials 467
§ 148. G. polynomials 473
Chapter IX. Numerical resolution of equations. Numerical
integration.
§ 149. Method of False Position. Regula Falsi ... 486
§ 150. The Newton Raphson method 489
§ 151. Method of Iteration 492
§ 152. Daniel Bernoulli s method 494
§ 153. The Ch in Vieta Horner method 496
§ 154. Root squaring method. Dandelin, Lobatchevsky,
Graeffe 503
xvii
§ 155. Numerical Integration 512
§ 156. Hardy and Weddle s formulae 516
§ 157. The Gauss Legendre method 517
§ 158. Tchebichef s formula 519
§ 159. Numerical integration of functions expanded into
a series of their differences 524
§ 160. Numerical resolution of differential equations . . 527
Chapter X. Functions of several independent variables.
§ 161. Functions of two variables 530
§ 162. Interpolation in a double entry table .... 532
§ 163. Functions of three variables 541
Chapter XI. Difference Equations.
§ 164. Genesis of difference equations 543
§ 165. Homogeneous linear difference equations, constant
coefficients 545
§ 166. Characteristic equations with multiple roots . . 549
§ 167. Negative roots 552
§ 168. Complex roots 554
§ 169. Complete linear equations of differences with
constant coefficients 557
§ 170. Determination of the particular solution in the
general case 564
§ 171. Method of the arbitrary constants 569
§ 172. Resolution of linear equations of differences by
aid of generating functions 572
§ 173. Homogeneous linear equations of the first order
with variable coefficients 576
§ 174. Laplace s method for solving linear homogeneous
difference equations with variable coefficients . . 579
§ 175. Complete linear equations of differences of the first
order with variable coefficients 583
§ 176. Reducible linear equations of differences with va¬
riable coefficients 584
§ 177. Linear equations of differences whose coefficients
are polynomials of x, solved by the method of
generating functions oo°
xviii
§ 178. Andre s method for solving difference equations . 587
§ 179. Sum equations which are reducible to equations of
differences 599
§ 180. Simultaneous linear equations of differences with
constant coefficients 600
Chapter XIII. Equations of Partial Differences.
§ 181. Introduction 604
§ 182. Resolution of linear equations of partial differences
with constant coefficients by Laplace s method of
generating functions 607
§ 183. Boole s symbolical method for solving equations of
partial differences 616
§ 184. Method of Fourier, Lagrange, and Ellis for solving
equations of partial differences 619
§ 185. Homogeneous linear equations of mixed differences 632
§ 186. Difference equations of three independent variables 633
§ 187. Differences equations of four independent variables 638
|
| any_adam_object | 1 |
| author | Jordan, Charles |
| author_facet | Jordan, Charles |
| author_role | aut |
| author_sort | Jordan, Charles |
| author_variant | c j cj |
| building | Verbundindex |
| bvnumber | BV002745341 |
| callnumber-first | Q - Science |
| callnumber-label | QA431 |
| callnumber-raw | QA431.J6 1965 |
| callnumber-search | QA431.J6 1965 |
| callnumber-sort | QA 3431 J6 41965 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 160 SK 580 |
| ctrlnum | (OCoLC)60944120 (DE-599)BVBBV002745341 |
| dewey-full | 517.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 517 - [Unassigned] |
| dewey-raw | 517.6 |
| dewey-search | 517.6 |
| dewey-sort | 3517.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 3. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T15:49:00Z |
| institution | BVB |
| language | English |
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| publisher | Chelsea Publ. Co. |
| record_format | marc |
| spelling | Jordan, Charles Verfasser aut Calculus of finite differences by Charles Jordan 3. ed. New York, NY Chelsea Publ. Co. 1965 XXI, 654 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Difference equations Interpolation Interpolation (DE-588)4162121-9 gnd rswk-swf Differenzenrechnung (DE-588)4149800-8 gnd rswk-swf Differenzengleichung (DE-588)4012264-5 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differenzenrechnung (DE-588)4149800-8 s DE-604 Differenzengleichung (DE-588)4012264-5 s Numerisches Verfahren (DE-588)4128130-5 s Differentialgleichung (DE-588)4012249-9 s Interpolation (DE-588)4162121-9 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=001755366&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 | Jordan, Charles Calculus of finite differences Difference equations Interpolation Interpolation (DE-588)4162121-9 gnd Differenzenrechnung (DE-588)4149800-8 gnd Differenzengleichung (DE-588)4012264-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4162121-9 (DE-588)4149800-8 (DE-588)4012264-5 (DE-588)4128130-5 (DE-588)4012249-9 |
| title | Calculus of finite differences |
| title_auth | Calculus of finite differences |
| title_exact_search | Calculus of finite differences |
| title_full | Calculus of finite differences by Charles Jordan |
| title_fullStr | Calculus of finite differences by Charles Jordan |
| title_full_unstemmed | Calculus of finite differences by Charles Jordan |
| title_short | Calculus of finite differences |
| title_sort | calculus of finite differences |
| topic | Difference equations Interpolation Interpolation (DE-588)4162121-9 gnd Differenzenrechnung (DE-588)4149800-8 gnd Differenzengleichung (DE-588)4012264-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Difference equations Interpolation Differenzenrechnung Differenzengleichung Numerisches Verfahren Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001755366&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT jordancharles calculusoffinitedifferences |