Calculus of finite differences:
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Chelsea Publ. Co.
1970
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| Edition: | 5. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 341 S. |
| ISBN: | 0828411212 |
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| adam_text | Titel: Calculus of finite differences
Autor: Boole, George
Jahr: 1970
CONTENTS. DIFFERENCE- AND SUM-CALCULUS. CHAPTER I. NATURE OF THE CALCULUS OF FINITE DIFFERENCES CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES Differences of Elementary Functions, 6. Expansion in factorials, 11. Generating Functions, 14. Laws and relations of E, A and 16. Secondary form of Maclaurin’s Theorem, 22. Herschel’s Theorem, 24. Miscellaneous Expansions, 25. Exercises, 28. CHAPTER III. ON INTERPOLATION, AND MECHANICAL QUADRATURE . Nature of the Problem, 33. Given values equidistant, 34. Not equidistant—Lagrange’s Method, 38. Gauss’ Method, 42. Cauchy’s Method, 43. Application to Statistics, 43. Areas of Curves, 46. Weddle’s rule, 48. Gauss’ Theorem on the best position of the given ordinates, 51. Laplace’s method of Quadratures, 53. Deferences on Interpolation, c. 55. Connexion between Gauss’ Theorem and Laplace’s Coefficients, 57. Exercises, 57. B. F. D. b
X CONTENTS. CHAPTER IV. page FINITE INTEGRATION, AND THE SUMMATION OF SERIES 62 Meaning of Integration, 62. Nature of the constant of Integration, 64. Definite and Indefinite Integrals, Go. Integrate forms and Summation of series—Factorials, 65. Inverse Factorials, 66. national and integral Functions, 68. Integrable Fractions, 70. Functions of the form a x p(x), 72. Miscellaneous Forms, 75. Bepeated Integration, 77. Conditions of extension of direct to inverse forms, 78. Periodical constants, 80. Analogy between the Integral and Sum-Calculus, 81. Eeferences, 83. Exercises, 83. CHAPTER V. THE APPROXIMATE SUMMATION OF SERIES 87 Development of 2, 87. Analogy with the methods adopted for the development of J, 87 (note). Division of the problem, 88. Development of 2 in powers of D (Euler-Maclaurin Formula), 89. Values of Bernoulli’s Numbers, 90. Applications, 91. Determination of Constant, 95. Development of 2 , 96. Development of 2 u x and 2“i t x in differences of a factor of u x , 99. Method of increasing the degree of approximation obtained by Maclaurin Theorem, 100. Expansion in inverse factorials, 102. Deferences, 103. Exercises, 103. CHAPTER VI. Bernoulli’s numbers, and factorial coefficients . 107 Various expressions for Bernoulli’s Numbers—De Moivre’s, 107. In terms of 2^, 109. Raabe’s (in factors), 109. As definite integrals, 110. Euler’s Numbers, 110. Bauer’s Theorem, 112. Factorial Coefficients, 113. Deferences, 116. Exercises, 117. CHAPTER VII. convergency and divergency of series 123 Definitions, 123. Case in which u x has aperiodic factor, 124. Cauchy’s Proposition, 126. First derived Criterion, 129. Supplemental Criteria—Bertrand’s Form, 132. De Morgan’s Form, 134. Third Form, 135. Theory of Degree, 136. Application of Tests to the Euler-Maclaurin Formula, 139. Order of Zeros, 139. Deferences, 140. Exercises, 140.
CONTENTS. XI CHAPTER VIII. PAGE EXACT THEOREMS . . . .145 Necessity for finding the limits of error in our expansions, 145. Remainder in the Generalized Form of Taylor’s Theorem, 146. Remainder in the Maclaurin Sum-Formula, 149. References, 152. Boole’s Limit of the Remainder of the Series for 2 k*, 154. DIFFERENCE- AND FUNCTIONAL EQUATIONS. CHAPTER IX. DIFFERENCE-EQUATIONS OF THE FIRST ORDER . 157 Definitions, 157. Genesis, 158. Existence of a complete Primitive, 160. Linear Equations of the First Order, 161. Difference-equations of the first order but not of the first degree—Clairault’s Form, 167. One variable absent, 167. Equations Homogeneous in u, 168.—Exercises, 169. CHAPTER X. GENERAL THEORY OF THE SOLUTIONS OF DIFFERENCE- AND DIFFERENTIAL EQUATIONS OF THE FIRST ORDER . 171 Difference-Equations—their solutions, 171. Derived Primitives, 172. Solutions derived from the Variation of a Constant, 174. Analogous method in Differential Equations, 177. Comparison between the solutions of Differential and Difference-Equations, 179. Associated primitives, 182. Possible non-existence of Complete Integral, 183. Detailed solution of u x = xAu x + (AmJ 2 , 185. Origin of singular solutions of Differential Equations, 189. Their analogues in Difference-Equations, 190. Remarks on the complete curves that satisfy a Differential Equation, 191. Anomalies of Singular Solutions, 193. Explanation of the same, 194. Principle of Continuity, 198. Recapitulation of the classes of solutions that a Difference-Equation may possess, 204. Exercises, 205. CHAPTER XI. LINEAR DIFFERENCE-EQUATIONS WITH CONSTANT COEFFICIENTS .... 208 Introductory remarks, 208. Solution of/ {E)u x = 0, 209. Solution of ƒ ( E)u x =X , 213. Examination of Symbolical methods, 215. Special forms of X, 218. Exercises, 219.
CONTENTS. xii CHAPTER XII. MISCELLANEOUS PROPOSITIONS AND EQUATIONS. SIMULTANEOUS EQUATIONS . Equations reducible to Linear Equations with Constant Coefficients, 221. Binomial Equations, 222. Depression of Linear Equations, 224. Generalization of the above, 225. Equations solved by performance of A , 228. Sylvester’s Forms, 229. Simultaneous Equations, 231. Exercises, 232. CHAPTER XIII. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. SYMBOLICAL AND GENERAL METHODS . Symbolical Methods, 230. Solution of Linear Difference-Equations in Series, 243. Finite Solution of Difference-Equations, 240. Binomial Equations, 248. Exercises, 203. CHAPTER XIV. MIXED AND PARTIAL DIFFERENCE-EQUATIONS Definitions, 264. Partial Difference-Equations, 266. Method of Generating Functions, 275. Mixed Difference-Equations, 277. Exercises, 289. CHAPTER XV. OF THE CALCULUS OF FUNCTIONS Definitions, 291, Direct Problems, 292. Periodical Functions, 298. Functional Equations, 301. Exercises, 312. CHAPTER XVI. GEOMETRICAL APPLICATIONS . Nature of the problems, 316. Miscellaneous instances, 317. Exercises, 325. PAGE 221 236 264 291 316 Answers to the Exercises . 326
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| adam_txt |
Titel: Calculus of finite differences
Autor: Boole, George
Jahr: 1970
CONTENTS. DIFFERENCE- AND SUM-CALCULUS. CHAPTER I. NATURE OF THE CALCULUS OF FINITE DIFFERENCES CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES Differences of Elementary Functions, 6. Expansion in factorials, 11. Generating Functions, 14. Laws and relations of E, A and 16. Secondary form of Maclaurin’s Theorem, 22. Herschel’s Theorem, 24. Miscellaneous Expansions, 25. Exercises, 28. CHAPTER III. ON INTERPOLATION, AND MECHANICAL QUADRATURE . Nature of the Problem, 33. Given values equidistant, 34. Not equidistant—Lagrange’s Method, 38. Gauss’ Method, 42. Cauchy’s Method, 43. Application to Statistics, 43. Areas of Curves, 46. Weddle’s rule, 48. Gauss’ Theorem on the best position of the given ordinates, 51. Laplace’s method of Quadratures, 53. Deferences on Interpolation, c. 55. Connexion between Gauss’ Theorem and Laplace’s Coefficients, 57. Exercises, 57. B. F. D. b
X CONTENTS. CHAPTER IV. page FINITE INTEGRATION, AND THE SUMMATION OF SERIES 62 Meaning of Integration, 62. Nature of the constant of Integration, 64. Definite and Indefinite Integrals, Go. Integrate forms and Summation of series—Factorials, 65. Inverse Factorials, 66. national and integral Functions, 68. Integrable Fractions, 70. Functions of the form a x p(x), 72. Miscellaneous Forms, 75. Bepeated Integration, 77. Conditions of extension of direct to inverse forms, 78. Periodical constants, 80. Analogy between the Integral and Sum-Calculus, 81. Eeferences, 83. Exercises, 83. CHAPTER V. THE APPROXIMATE SUMMATION OF SERIES 87 Development of 2, 87. Analogy with the methods adopted for the development of J, 87 (note). Division of the problem, 88. Development of 2 in powers of D (Euler-Maclaurin Formula), 89. Values of Bernoulli’s Numbers, 90. Applications, 91. Determination of Constant, 95. Development of 2", 96. Development of 2 u x and 2“i t x in differences of a factor of u x , 99. Method of increasing the degree of approximation obtained by Maclaurin Theorem, 100. Expansion in inverse factorials, 102. Deferences, 103. Exercises, 103. CHAPTER VI. Bernoulli’s numbers, and factorial coefficients . 107 Various expressions for Bernoulli’s Numbers—De Moivre’s, 107. In terms of 2^, 109. Raabe’s (in factors), 109. As definite integrals, 110. Euler’s Numbers, 110. Bauer’s Theorem, 112. Factorial Coefficients, 113. Deferences, 116. Exercises, 117. CHAPTER VII. convergency and divergency of series 123 Definitions, 123. Case in which u x has aperiodic factor, 124. Cauchy’s Proposition, 126. First derived Criterion, 129. Supplemental Criteria—Bertrand’s Form, 132. De Morgan’s Form, 134. Third Form, 135. Theory of Degree, 136. Application of Tests to the Euler-Maclaurin Formula, 139. Order of Zeros, 139. Deferences, 140. Exercises, 140.
CONTENTS. XI CHAPTER VIII. PAGE EXACT THEOREMS . . . .145 Necessity for finding the limits of error in our expansions, 145. Remainder in the Generalized Form of Taylor’s Theorem, 146. Remainder in the Maclaurin Sum-Formula, 149. References, 152. Boole’s Limit of the Remainder of the Series for 2 k*, 154. DIFFERENCE- AND FUNCTIONAL EQUATIONS. CHAPTER IX. DIFFERENCE-EQUATIONS OF THE FIRST ORDER . 157 Definitions, 157. Genesis, 158. Existence of a complete Primitive, 160. Linear Equations of the First Order, 161. Difference-equations of the first order but not of the first degree—Clairault’s Form, 167. One variable absent, 167. Equations Homogeneous in u, 168.—Exercises, 169. CHAPTER X. GENERAL THEORY OF THE SOLUTIONS OF DIFFERENCE- AND DIFFERENTIAL EQUATIONS OF THE FIRST ORDER . 171 Difference-Equations—their solutions, 171. Derived Primitives, 172. Solutions derived from the Variation of a Constant, 174. Analogous method in Differential Equations, 177. Comparison between the solutions of Differential and Difference-Equations, 179. Associated primitives, 182. Possible non-existence of Complete Integral, 183. Detailed solution of u x = xAu x + (AmJ 2 , 185. Origin of singular solutions of Differential Equations, 189. Their analogues in Difference-Equations, 190. Remarks on the complete curves that satisfy a Differential Equation, 191. Anomalies of Singular Solutions, 193. Explanation of the same, 194. Principle of Continuity, 198. Recapitulation of the classes of solutions that a Difference-Equation may possess, 204. Exercises, 205. CHAPTER XI. LINEAR DIFFERENCE-EQUATIONS WITH CONSTANT COEFFICIENTS . 208 Introductory remarks, 208. Solution of/ {E)u x = 0, 209. Solution of ƒ ( E)u x =X , 213. Examination of Symbolical methods, 215. Special forms of X, 218. Exercises, 219.
CONTENTS. xii CHAPTER XII. MISCELLANEOUS PROPOSITIONS AND EQUATIONS. SIMULTANEOUS EQUATIONS . Equations reducible to Linear Equations with Constant Coefficients, 221. Binomial Equations, 222. Depression of Linear Equations, 224. Generalization of the above, 225. Equations solved by performance of A", 228. Sylvester’s Forms, 229. Simultaneous Equations, 231. Exercises, 232. CHAPTER XIII. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. SYMBOLICAL AND GENERAL METHODS . Symbolical Methods, 230. Solution of Linear Difference-Equations in Series, 243. Finite Solution of Difference-Equations, 240. Binomial Equations, 248. Exercises, 203. CHAPTER XIV. MIXED AND PARTIAL DIFFERENCE-EQUATIONS Definitions, 264. Partial Difference-Equations, 266. Method of Generating Functions, 275. Mixed Difference-Equations, 277. Exercises, 289. CHAPTER XV. OF THE CALCULUS OF FUNCTIONS Definitions, 291, Direct Problems, 292. Periodical Functions, 298. Functional Equations, 301. Exercises, 312. CHAPTER XVI. GEOMETRICAL APPLICATIONS . Nature of the problems, 316. Miscellaneous instances, 317. Exercises, 325. PAGE 221 236 264 291 316 Answers to the Exercises . 326 |
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| spelling | Boole, George 1815-1864 Verfasser (DE-588)118661655 aut Calculus of finite differences by George Boole 5. ed. New York, NY Chelsea Publ. Co. 1970 341 S. txt rdacontent n rdamedia nc rdacarrier Calcul différentiel calcul différentiel inriac fonction Bernoulli inriac méthode différence finie inriac Difference equations Functional equations Differenzenrechnung (DE-588)4149800-8 gnd rswk-swf Differenzenrechnung (DE-588)4149800-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015150601&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boole, George 1815-1864 Calculus of finite differences Calcul différentiel calcul différentiel inriac fonction Bernoulli inriac méthode différence finie inriac Difference equations Functional equations Differenzenrechnung (DE-588)4149800-8 gnd |
| subject_GND | (DE-588)4149800-8 |
| title | Calculus of finite differences |
| title_auth | Calculus of finite differences |
| title_exact_search | Calculus of finite differences |
| title_exact_search_txtP | Calculus of finite differences |
| title_full | Calculus of finite differences by George Boole |
| title_fullStr | Calculus of finite differences by George Boole |
| title_full_unstemmed | Calculus of finite differences by George Boole |
| title_short | Calculus of finite differences |
| title_sort | calculus of finite differences |
| topic | Calcul différentiel calcul différentiel inriac fonction Bernoulli inriac méthode différence finie inriac Difference equations Functional equations Differenzenrechnung (DE-588)4149800-8 gnd |
| topic_facet | Calcul différentiel calcul différentiel fonction Bernoulli méthode différence finie Difference equations Functional equations Differenzenrechnung |
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