Operational mathematics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Tokyo [u.a.]
McGraw-Hill Kogakusha
1972
|
| Ausgabe: | 3. ed., internat. student ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 481 S. |
Internformat
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Datensatz im Suchindex
| _version_ | 1804135831250665472 |
|---|---|
| adam_text | Contents
Preface xi
Chapter 1. The Laplace Transformation ]
1. Introduction 1
2. Definition of the Laplace Transformation 3
3. Sectionally Continuous Functions. Exponential Order 5
4. Transforms of Derivatives 7
5. Examples. The Gamma Function 10
6. The Inverse Transform 14
7. A Theorem on Substitution 16
8. The Use of Pa rtial Fractions (Table 1) 17
9. The Solution of Simple Differential Equations 20
10. Generation of the Transformation 24
Chapter 2. Further Properties of the Transformation 27
11. Translation of F(f) 27
12. Step Functions 29
13. The Impulse Symbol 5(f t0) 33
14. Integrals Containing a Parameter 39
15. Improper Integrals 41
16. Convolution 43
17. Properties of Convolution 46
18. Differential and Integral Equations SO
19. Derivatives of Transforms 54
20. Series of Transforms 57
21. Differential Equations with Variable Coefficients 61
22. Integration of Transforms 65
v
vi CONTENTS
23. Periodic Functions 66
24. Partial Fractions 70
25. Repeated Linear Factors 73
26. Quadratic Factors 75
27. Tables of Operations and Transforms 78
Chapter 3. Elementary Applications 85
28. Free Vibrations of a Mass on a Spring 85
29. Forced Vibrations without Damping 88
30. Resonance 91
31. Forced Vibrations with Damping 95
32. A Vibration Absorber 96
33. Electric Circuits 102
34. Evaluation of Integrals 106
35. Exponential and Cosine integral Functions 110
36. Static Deflection of Beams 113
37. The Tautochrone 115
38. Servomechanisms 117
39. Mortality of Equipment 119
Chapter 4. Problems in Partial Differential Equations 123
40. The Wave Equation 123
41. Displacements in a Long String 126
42. A Long String under Its Weight 130
43. The Long String Initially Displaced 133
44. A Bar with a Prescribed Force on One End 135
45. Equations of Diffusion 143
46. Temperatures in a Semi infinite Solid 145
47. Prescribed Surface Temperature 146
48. Temperatures in a Slab 151
49. A Bar with Variable End Temperature 153
50. A Cooling Fin or Evaporation Plate 153
51. Temperatures in a Composite Solid 155
52. Observations on the Method 159
Chapter S. Functions of a Complex Variable 162
53. Complex Numbers 162
54. Analytic Functions 163
55. Exponential and Trigonometric Functions 166
56. Contour Integrals 168
57. Integral Theorems 170
58. Power Series 171
59. Singular Points and Residues 173
CONTENTS vii
60. Branches of Multiple valued Functions 177
61. Analytic Continuation 179
62. Improper Cauchy Integrals 181
Chapter 6. The Inversion Integral 186
63. Analytic Transforms 186
64. Permanence of Forms 188
65. Order Properties of Transforms 189
66. The Inversion Integral 193
67. Conditions on/(s) 195
68. Conditions on F(t) 198
69. Uniqueness of Inverse Transforms 201
70. Derivatives of the Inversion Integral 202
71. Representation by Series of Residues 206
72. Residues at Poles 208
73. Validity of the Representation by Series 210
74. Alterations of the Inversion Integral 213
Chapter 7. Problems in Heat Conduction 219
75. Temperatures in a Bar with Ends at Fixed Temperatures 220
76. The Solution Established 222
77. The Series Form Established 224
78. Properties of the Temperature Function 226
79. Uniqueness of the Solution 228
80. Arbitrary End Temperatures 230
81. Special End Temperatures 232
82. Arbitrary Initial Temperatures 237
83. Temperatures in a Cylinder 240
84. Evaporation from a Thick Slab 245
85. Duhamel s Formula 247
Chapter 8. Problems in Mechanical Vibrations 253
86. A Bar with a Constant Force on One End 253
87. Another Form of the Solution 256
88. Resonance in the Bar with a Fixed End 258
89. Verification of Solutions 259
90. Free Vibrations of a String 264
91. Resonance in a Bar with a Mass Attached 266
92. Transverse Vibrations of Beams 268
93. Duhamel s Formula for Vibration Problems 270
Chapter* Generalized Fourier Series 276
94. Self adjoint Differential Equations 277
viii CONTENTS
95. Green s Functions 278
96. Construction of Green s Function 281
97. Orthogonal Sets of Functions 283
98. Eigenvalue Problems 288
99. A Representation Theorem 291
100. The Reduced Sturm Liouville System 292
101. A Related Boundary Value Problem 293
102. The Transform y(x,s) 294
103. Existence of Eigenvalues 296
104. The Generalized Fourier Series 299
105. Steady Temperatures in a Wall 305
106. Verification of the Solution 308
107. Singular Eigenvalue Problems 309
Chapter 10. General Integral Transforms 317
108. Linear Integral Transformations 317
109. Kernel product Convolution Properties 319
110. Example 320
111. Sturm Liouville Transforms 325
112. Inverse Transforms 327
113. Further Properties 329
114. Transforms of Certain Functions 332
115. Example of Sturm Liouville Transformations 333
116. Singular Cases 336
11 /. A Problem in Steady Temperatures 341
118. Other Boundary Value Problems 343
Chapter 11. Finite Fourier Transforms 348
119. Finite Fourier Sine Transforms 348
120. Other Properties of S. 350
121. Finite Cosine Transforms 354
122. Tables of Finite Fourier Transforms 356
123. Joint Properties of C, and S, 357
124. Potential in a Slot 360
125. Successive Transformations 362
126. A Modified Sine Transformation 368
127. Generalized Cosine Transforms 370
128. A Generalized Sine Transform 372
129. Finite Exponential Transforms E,{F 376
130. Other Properties off, 378
Chapter 12. Exponential Fourier Transforms 383
131. The Transformation E,{F] 383
CONTENTS jx
132. The Inverse Transformation 385
133. Other Properties of £, 388
134. The Convolution Integral for £„ 391
135. Convolution Theorem 393
136. Tables of Transforms 396
137. Boundary Value Problems 397
Chapter 13. Fourier Transforms on the Half Line 401
138. Fourier Sine Transforms/,(*) 401
139. Fourier Cosine Transforms fc(oi) 404
140. Further Properties of S, and C, 405
141. Convolution Properties 406
142. Tables of Sine and Cosine Transforms 407
143. Steady Temperatures in a Quadrant 408
144. Deflections in an Elastic Plate 410
145. A Modified Fourier Transformation Tr 414
146. Convolution for Tt 416
147. Surface Heat Transfer 418
Chapter 14. Hankel Transforms 420
148. Introduction 420
149. Finite Hankel Transformations 421
150. Inversion of Hnj 423
151. Modified Finite Transformations //„, 424
152. A Boundary Value Problem 426
153. Nonsingular Hankel Transformations 431
154. Hankel Transformations Hnl on the Half Line (x 0) 432
155. Further Properties of //„, 434
156. Tables of Transforms //„ {F J 436
157. Axially Symmetric Heat Source 437
Chapter 15. Legendre and Other Integral Transforms 441
158. The Legendre Transformation Tn on the Interval ( 1,1) 442
159. Further Properties of Tn 444
160. Legendre Transforms on the Interval (0,1) 446
161. Dirichlet Problems for the Sphere 448
162. Laguerre Transforms 452
163. Mellin Transforms 453
Bibliography 456
Appendixes
Appendix A Tables of Laplace Transforms 4SX
x CONTENTS
Table A.I Operations 458
Table A.2 Laplace Transforms 4S9
Appendix B Tables of Finite Fourier Transforms 467
Table B. 1 Finite Sine Transforms 467
Table B.2 Finite Cosine Transforms 469
Appendix C Table of Exponential Fourier Transforms 471
Appendix D Tables of Fourier Sine and Cosine Transforms 473
Table D.I Sine Transforms on the Half Line 473
Table D.2 Cosine Transforms on the Half Line 475
Index 477
|
| adam_txt |
Contents
Preface xi
Chapter 1. The Laplace Transformation ]
1. Introduction 1
2. Definition of the Laplace Transformation 3
3. Sectionally Continuous Functions. Exponential Order 5
4. Transforms of Derivatives 7
5. Examples. The Gamma Function 10
6. The Inverse Transform 14
7. A Theorem on Substitution 16
8. The Use of Pa'rtial Fractions (Table 1) 17
9. The Solution of Simple Differential Equations 20
10. Generation of the Transformation 24
Chapter 2. Further Properties of the Transformation 27
11. Translation of F(f) 27
12. Step Functions 29
13. The Impulse Symbol 5(f t0) 33
14. Integrals Containing a Parameter 39
15. Improper Integrals 41
16. Convolution 43
17. Properties of Convolution 46
18. Differential and Integral Equations SO
19. Derivatives of Transforms 54
20. Series of Transforms 57
21. Differential Equations with Variable Coefficients 61
22. Integration of Transforms 65
v
vi CONTENTS
23. Periodic Functions 66
24. Partial Fractions 70
25. Repeated Linear Factors 73
26. Quadratic Factors 75
27. Tables of Operations and Transforms 78
Chapter 3. Elementary Applications 85
28. Free Vibrations of a Mass on a Spring 85
29. Forced Vibrations without Damping 88
30. Resonance 91
31. Forced Vibrations with Damping 95
32. A Vibration Absorber 96
33. Electric Circuits 102
34. Evaluation of Integrals 106
35. Exponential and Cosine integral Functions 110
36. Static Deflection of Beams 113
37. The Tautochrone 115
38. Servomechanisms 117
39. Mortality of Equipment 119
Chapter 4. Problems in Partial Differential Equations 123
40. The Wave Equation 123
41. Displacements in a Long String 126
42. A Long String under Its Weight 130
43. The Long String Initially Displaced 133
44. A Bar with a Prescribed Force on One End 135
45. Equations of Diffusion 143
46. Temperatures in a Semi infinite Solid 145
47. Prescribed Surface Temperature 146
48. Temperatures in a Slab 151
49. A Bar with Variable End Temperature 153
50. A Cooling Fin or Evaporation Plate 153
51. Temperatures in a Composite Solid 155
52. Observations on the Method 159
Chapter S. Functions of a Complex Variable 162
53. Complex Numbers 162
54. Analytic Functions 163
55. Exponential and Trigonometric Functions 166
56. Contour Integrals 168
57. Integral Theorems 170
58. Power Series 171
59. Singular Points and Residues 173
CONTENTS vii
60. Branches of Multiple valued Functions 177
61. Analytic Continuation 179
62. Improper Cauchy Integrals 181
Chapter 6. The Inversion Integral 186
63. Analytic Transforms 186
64. Permanence of Forms 188
65. Order Properties of Transforms 189
66. The Inversion Integral 193
67. Conditions on/(s) 195
68. Conditions on F(t) 198
69. Uniqueness of Inverse Transforms 201
70. Derivatives of the Inversion Integral 202
71. Representation by Series of Residues 206
72. Residues at Poles 208
73. Validity of the Representation by Series 210
74. Alterations of the Inversion Integral 213
Chapter 7. Problems in Heat Conduction 219
75. Temperatures in a Bar with Ends at Fixed Temperatures 220
76. The Solution Established 222
77. The Series Form Established 224
78. Properties of the Temperature Function 226
79. Uniqueness of the Solution 228
80. Arbitrary End Temperatures 230
81. Special End Temperatures 232
82. Arbitrary Initial Temperatures 237
83. Temperatures in a Cylinder 240
84. Evaporation from a Thick Slab 245
85. Duhamel's Formula 247
Chapter 8. Problems in Mechanical Vibrations 253
86. A Bar with a Constant Force on One End 253
87. Another Form of the Solution 256
88. Resonance in the Bar with a Fixed End 258
89. Verification of Solutions 259
90. Free Vibrations of a String 264
91. Resonance in a Bar with a Mass Attached 266
92. Transverse Vibrations of Beams 268
93. Duhamel's Formula for Vibration Problems 270
Chapter* Generalized Fourier Series 276
94. Self adjoint Differential Equations 277
viii CONTENTS
95. Green's Functions 278
96. Construction of Green's Function 281
97. Orthogonal Sets of Functions 283
98. Eigenvalue Problems 288
99. A Representation Theorem 291
100. The Reduced Sturm Liouville System 292
101. A Related Boundary Value Problem 293
102. The Transform y(x,s) 294
103. Existence of Eigenvalues 296
104. The Generalized Fourier Series 299
105. Steady Temperatures in a Wall 305
106. Verification of the Solution 308
107. Singular Eigenvalue Problems 309
Chapter 10. General Integral Transforms 317
108. Linear Integral Transformations 317
109. Kernel product Convolution Properties 319
110. Example 320
111. Sturm Liouville Transforms 325
112. Inverse Transforms 327
113. Further Properties 329
114. Transforms of Certain Functions 332
115. Example of Sturm Liouville Transformations 333
116. Singular Cases 336
11 /. A Problem in Steady Temperatures 341
118. Other Boundary Value Problems 343
Chapter 11. Finite Fourier Transforms 348
119. Finite Fourier Sine Transforms 348
120. Other Properties of S. 350
121. Finite Cosine Transforms 354
122. Tables of Finite Fourier Transforms 356
123. Joint Properties of C, and S, 357
124. Potential in a Slot 360
125. Successive Transformations 362
126. A Modified Sine Transformation 368
127. Generalized Cosine Transforms 370
128. A Generalized Sine Transform 372
129. Finite Exponential Transforms E,{F\ 376
130. Other Properties off, 378
Chapter 12. Exponential Fourier Transforms 383
131. The Transformation E,{F] 383
CONTENTS jx
132. The Inverse Transformation 385
133. Other Properties of £, 388
134. The Convolution Integral for £„ 391
135. Convolution Theorem 393
136. Tables of Transforms 396
137. Boundary Value Problems 397
Chapter 13. Fourier Transforms on the Half Line 401
138. Fourier Sine Transforms/,(*) 401
139. Fourier Cosine Transforms fc(oi) 404
140. Further Properties of S, and C, 405
141. Convolution Properties 406
142. Tables of Sine and Cosine Transforms 407
143. Steady Temperatures in a Quadrant 408
144. Deflections in an Elastic Plate 410
145. A Modified Fourier Transformation Tr 414
146. Convolution for Tt 416
147. Surface Heat Transfer 418
Chapter 14. Hankel Transforms 420
148. Introduction 420
149. Finite Hankel Transformations 421
150. Inversion of Hnj 423
151. Modified Finite Transformations //„, 424
152. A Boundary Value Problem 426
153. Nonsingular Hankel Transformations 431
154. Hankel Transformations Hnl on the Half Line (x 0) 432
155. Further Properties of //„, 434
156. Tables of Transforms //„ {F J 436
157. Axially Symmetric Heat Source 437
Chapter 15. Legendre and Other Integral Transforms 441
158. The Legendre Transformation Tn on the Interval ( 1,1) 442
159. Further Properties of Tn 444
160. Legendre Transforms on the Interval (0,1) 446
161. Dirichlet Problems for the Sphere 448
162. Laguerre Transforms 452
163. Mellin Transforms 453
Bibliography 456
Appendixes
Appendix A Tables of Laplace Transforms 4SX
x CONTENTS
Table A.I Operations 458
Table A.2 Laplace Transforms 4S9
Appendix B Tables of Finite Fourier Transforms 467
Table B. 1 Finite Sine Transforms 467
Table B.2 Finite Cosine Transforms 469
Appendix C Table of Exponential Fourier Transforms 471
Appendix D Tables of Fourier Sine and Cosine Transforms 473
Table D.I Sine Transforms on the Half Line 473
Table D.2 Cosine Transforms on the Half Line 475
Index 477 |
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| author | Churchill, Ruel Vance 1899-1987 |
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| author_facet | Churchill, Ruel Vance 1899-1987 |
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| author_sort | Churchill, Ruel Vance 1899-1987 |
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| building | Verbundindex |
| bvnumber | BV021888776 |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)258363848 (DE-599)BVBBV021888776 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 3. ed., internat. student ed. |
| format | Book |
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| id | DE-604.BV021888776 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:04:05Z |
| indexdate | 2024-07-09T20:46:45Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015104005 |
| oclc_num | 258363848 |
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| owner_facet | DE-706 |
| physical | 481 S. |
| publishDate | 1972 |
| publishDateSearch | 1972 |
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| publisher | McGraw-Hill Kogakusha |
| record_format | marc |
| spelling | Churchill, Ruel Vance 1899-1987 Verfasser (DE-588)115367608 aut Operational mathematics 3. ed., internat. student ed. Tokyo [u.a.] McGraw-Hill Kogakusha 1972 481 S. txt rdacontent n rdamedia nc rdacarrier Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd rswk-swf Operatortheorie (DE-588)4075665-8 gnd rswk-swf Laplace-Transformation (DE-588)4034577-4 gnd rswk-swf Fourier-Transformation (DE-588)4018014-1 gnd rswk-swf Integraltransformation (DE-588)4027235-7 gnd rswk-swf Fourier-Transformation (DE-588)4018014-1 s DE-604 Laplace-Transformation (DE-588)4034577-4 s Integraltransformation (DE-588)4027235-7 s Lineare partielle Differentialgleichung (DE-588)4167708-0 s 1\p DE-604 Operatortheorie (DE-588)4075665-8 s 2\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015104005&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 |
| spellingShingle | Churchill, Ruel Vance 1899-1987 Operational mathematics Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd Operatortheorie (DE-588)4075665-8 gnd Laplace-Transformation (DE-588)4034577-4 gnd Fourier-Transformation (DE-588)4018014-1 gnd Integraltransformation (DE-588)4027235-7 gnd |
| subject_GND | (DE-588)4167708-0 (DE-588)4075665-8 (DE-588)4034577-4 (DE-588)4018014-1 (DE-588)4027235-7 |
| title | Operational mathematics |
| title_auth | Operational mathematics |
| title_exact_search | Operational mathematics |
| title_exact_search_txtP | Operational mathematics |
| title_full | Operational mathematics |
| title_fullStr | Operational mathematics |
| title_full_unstemmed | Operational mathematics |
| title_short | Operational mathematics |
| title_sort | operational mathematics |
| topic | Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd Operatortheorie (DE-588)4075665-8 gnd Laplace-Transformation (DE-588)4034577-4 gnd Fourier-Transformation (DE-588)4018014-1 gnd Integraltransformation (DE-588)4027235-7 gnd |
| topic_facet | Lineare partielle Differentialgleichung Operatortheorie Laplace-Transformation Fourier-Transformation Integraltransformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015104005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT churchillruelvance operationalmathematics |