Operational calculus: based on the 2-sided Laplace integral
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Chelsea Publ.
1987
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| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 409 S. graph. Darst. |
| ISBN: | 0828403279 |
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| 245 | 1 | 0 | |a Operational calculus |b based on the 2-sided Laplace integral |c by Balth. van der Pol and H. Bremmer |
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Record in the Search Index
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| adam_text | Titel: Operational calculus
Autor: Pol, Balthasar van der
Jahr: 1987
CONTENTS I GENERAL INTRODUCTION 1. History of the operational calculus page 1 2. The operational calculus based on the Laplace transform 2 3. Survey of the subject-matter 5 II THE FOURIER INTEGRAL AS BASIS OF THE OPERATIONAL CALCULUS 1. The Fourier integral 7 2. A pair of integrals equivalent to the Fourier identity 9 3. Parallel displacement of the path of integration in the integral (56) 11 4. Rotation of the path of integration in the integral (11 6) 12 5. Strip of convergence of the Laplace integral 13 6. The language of the operational calculus 18 7. Operational calculus based on one-sided Laplace integrals 20 III ELEMENTARY OPERATIONAL IMAGES 1. Introduction 22 2. Images of polynomials; the T-function 23 3. Images of simple exponential functions 26 4. Images of complicated exponential functions 27 5. Images of trigonometric functions 29 6. Images of logarithmic functions 30 7. Well-known integrals of the Laplace type 31 IV ELEMENTARY RULES 1. Introduction 33 2. The similarity rule 33 3. The shift rule 34 4. The attenuation rule 38 5. The composition product 39 6. Repeated composition product 43 7. The image of the product of two originals 47 8. The differentiation rule 48 9. The integration rule 51 10. Rules for multiplication by t n 53 11. Rules for division by t, and related integrals 53 12. Rules for the treatment of correlation functions 55 V THE DELTA OR IMPULSE FUNCTION 1. Introduction 56 2. The unit function 56 3. The 5-function as derivative of the unit function 59
X CONTENTS 4. History of the impulse function page 62 6. The sifting integral as a Stieltjes integral 66 6. The image of the ¿-function 68 7. Functions approximating the ¿-function 70 8. Applications of the ¿-function 74 9. Series of impulse functions 78 10. Derivatives of the ¿-function 80 11. Note on the theory of distributions 84 VI QUESTIONS CONCERNING THE CONVERGENCE OE THE DEFINITION INTEGRAL 1. Introduction 85 2. Extensions of the definition integral 85 3. The operational treatment of some special series 87 4. The strip of convergence of operational relations 91 5. Particular cases of the strip of convergence 94 6. Absolute convergence 96 7. Uniform convergence 97 8. Summable series 100 9. Summable integrals 102 10. The behaviour of the definition integral on the boundaries of the strip of convergence 104 11. The inversion integral 106 12. Adjacent strips of convergence for the same image 109 13. Operational relations having a line of convergence 113 14. Symmetry between the integrals of definition and inversion ] 15 15. Uniqueness of the operational relations 117 VII ASYMPTOTIC RELATIONS AND OPERATIONAL TRANS- POSITION OF SERIES 1. Introduction 121 2. Abel and Tauber theorems 122 3. Abel theorems ] 24 4. Real Tauber theorems 128 5. Complex Tauber theorems 130 6. Operational equalities 133 7. Asymptotic series 134 8. Operational transposition of power series in p _1 and in t 136 9. Transposition of series with ascending powers of p 139 10. Expansion in rational fractions (Heaviside’s expansion theorem II) 142 11. Transposition of other series 147 12. A real inversion formula 148
CONTENTS XI VIII LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 1. Introduction page 152 2. Inhomogeneous equations with the unit function at the right 152 3. Inhomogeneous equations with arbitrary right-hand member 155 4. Differential equations with boundary conditions 157 5. Admittances and impedances 161 6. Transient phenomena 167 7. Time impedances 170 8. Series for small values of t. Heaviside’s first expansion theorem 171 9. Classification of admittances 173 IX SIMULTANEOUS LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS; ELECTRIC-CIRCUIT THEORY 1. Introduction 175 2. Equations of electric-circuit theory 175 3. Transposition of the circuit equations; mutual impedances and admittances 177 4. Time admittances of electric circuits 180 5. Applications of the general circuit theory 182 6. Circuit equations with initial conditions 184 7. Ladder networks; filters 185 8. Lattice networks 194 X LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 1. Introduction 200 2. Laguerre polynomials 201 3. Hermite polynomials 204 4. Bessel functions (operational relations) 207 5. Bessel functions (applications) 214 6. Legendre functions (operational relations) 220 7. Legendre functions (applications) 224 8. Hypergeometric functions 225 XI OPERATIONAL RULES OF MORE COMPLICATED CHARACTER 1. Introduction 232 2. Rules obtained when p is replaced by a function of p 232 3. Rules obtained when t is replaced by a function of t 237 4. The exponential transformation: i-*-e‘ 238 6. Exponential operational relations for power series, in particular hypergeometric series 243 Rules concerning series expansions 249 6 .
CONTENTS xii 7. Equalities in connexion with a single operational relation page 253 8. Equalities in connexion with a pair of simultaneous relations (exchange identity) 254 XII STEP FUNCTIONS AND OTHER DISCONTINUOUS FUNCTIONS 1. Introduction 257 2. Operational relations involving step functions jumping at integral values of t 258 3. The saw-tooth function 260 4. Arithmetic functions in connexion with ^-functions 263 5. Aritlimetic functions in connexion with Dirichlet series 266 6. Step functions of argument equal to the summation variable in a series 272 7. Contragrade series 274 Xin DIFFERENCE EQUATIONS 1. Introduction 277 2. Difference equation for the ‘sum’ 278 3. General linear difference equations with constant coefficients 281 4. Linear difference equations with variable coefficients 285 5. Connexion between differential and difference equations 287 6. Operational construction of difference equations 289 XIV INTEGRAL EQUATIONS 1. Introduction 292 2. The integral equation for the moving average 293 3. Integral equations of the first kind with difference kernel 300 4. Integral equations of the first kind with kernel reducible to a difference kernel 305 5. Integral equations of the second kind with a difference kernel 307 6. Homogeneous integral equations 310 7. The operational construction of integral equations 312 8. Note on the operational interpretation of the Wiener-Hopf technique 313 XV PARTIAL DIFFERENTIAL EQUATIONS IN THE OPERATIONAL CALCULUS OF ONE VARIABLE 1. Introduction 314 2. Homogeneous linear partial differential equations (general solutions) 317 3. Homogeneous partial differential equations with boundary conditions 322 4. Quasi-stat-ionary theory of electric cables 325 5. Inhomogeneous partial differential equations 330
CONTENTS au XVI SIMULTANEOUS OPERATIONAL CALCULUS 1. Introduction page 334 2. General theory 334 3. Second-order differential equations of the hyperbolic type with constant coefficients and two variables 344 4. Hyperbolic differential equations of the second order in more than two independent variables 352 5. Elliptic differential equations 355 6. Simultaneous partial differential equations 361 7. Partial difference equations 365 xvn ‘grammar’ 373 xvhi ‘dictionary’ 383 LIST OF AUTHORS QUOTED 411 GENERAL INDEX 413
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| any_adam_object | 1 |
| author | Pol, Balthasar van der Bremmer, Hendricus 1904-1996 |
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| dewey-full | 515.7/2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:48:28Z |
| institution | BVB |
| isbn | 0828403279 |
| language | English |
| lccn | 86071694 |
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| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XIII, 409 S. graph. Darst. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Chelsea Publ. |
| record_format | marc |
| spelling | Pol, Balthasar van der Verfasser aut Operational calculus based on the 2-sided Laplace integral by Balth. van der Pol and H. Bremmer 3. ed. New York, NY Chelsea Publ. 1987 XIII, 409 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Calculus, Operational Operatorenrechnung (DE-588)4389717-4 gnd rswk-swf Laplace-Transformation (DE-588)4034577-4 gnd rswk-swf Laplace-Transformation (DE-588)4034577-4 s DE-604 Operatorenrechnung (DE-588)4389717-4 s Bremmer, Hendricus 1904-1996 Verfasser (DE-588)123226929 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004190933&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pol, Balthasar van der Bremmer, Hendricus 1904-1996 Operational calculus based on the 2-sided Laplace integral Calculus, Operational Operatorenrechnung (DE-588)4389717-4 gnd Laplace-Transformation (DE-588)4034577-4 gnd |
| subject_GND | (DE-588)4389717-4 (DE-588)4034577-4 |
| title | Operational calculus based on the 2-sided Laplace integral |
| title_auth | Operational calculus based on the 2-sided Laplace integral |
| title_exact_search | Operational calculus based on the 2-sided Laplace integral |
| title_full | Operational calculus based on the 2-sided Laplace integral by Balth. van der Pol and H. Bremmer |
| title_fullStr | Operational calculus based on the 2-sided Laplace integral by Balth. van der Pol and H. Bremmer |
| title_full_unstemmed | Operational calculus based on the 2-sided Laplace integral by Balth. van der Pol and H. Bremmer |
| title_short | Operational calculus |
| title_sort | operational calculus based on the 2 sided laplace integral |
| title_sub | based on the 2-sided Laplace integral |
| topic | Calculus, Operational Operatorenrechnung (DE-588)4389717-4 gnd Laplace-Transformation (DE-588)4034577-4 gnd |
| topic_facet | Calculus, Operational Operatorenrechnung Laplace-Transformation |
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