《Operational Calculus Based on The Two-Sided Laplace Integral Second Edition》

ⅠGENERAL INTRODUCTION1

1．History of the operational calculus1

2．The operational calculus based on the Laplace transform2

3．Survey of the subject-matter5

ⅡTHE FOURIER INTEGRAL AS BASIS OP THE OPERATIONAL CALCULUS7

1．The Fourier integral7

2．A pair of integrals equivalent to the Fourier identity9

3．Parallel displacement of the path of integration in the integral(5b)11

4．Rotation of the path of integration in the integral(11b)12

5．Strip of convergence of the Laplace integral13

6．The language of the operational calculus18

7．Operational calculus based on one-sided Laplace integrals20

ⅢELEMENTARY OPERATIONAL IMAGES22

1．Introduction22

2．Images of polynomials；the Γ-function23

3．Images of simple exponential functions26

4．Images of complicated exponential functions27

5．Images of trigonometric functions29

6．Images of logarithmic functions30

7．Well-known integrals of the Laplace type31

ⅣELEMENTARY RULES33

1．Introduction33

2．The similarity rule33

3．The shift rule34

4．The attenuation rule38

5．The composition product39

6．Repeated composition product43

7．The image of the product of two originals47

8．The differentiation rule48

9．The integration rule51

10．Rules for multiplication by tn53

11．Rules for division by t，and related integrals53

12．Rules for the treatment of correlation functions55

ⅤTHE DELTA OR IMPULSE FUNCTION56

1．Introduction56

2．The unit function56

3．The δ-function as derivative of the unit function59

4．History of the impulse function62

5．The sifting integral as a Stieltjes integral66

6．The image of the δ-function68

7．Functions approximating the δ-function70

8．Applications of the δ-function74

9．Series of impulse functions78

10．Derivatives of the δ-function80

11．Note on the theory of distributions84

ⅥQUESTIONS CONCERNING THE CONVERGENCE OF THE DEFINITION INTEGRAL85

1．Introduction85

2．Extensions of the definition integral85

3．The operational treatment of some special series87

4．The strip of convergence of operational relations91

5．Particular cases of the strip of convergence94

6．Absolute convergence96

7．Uniform convergence97

8．Summable series100

9．Summable integrals102

10．The behaviour of the definition integral on the boundaries of the strip of convergence104

11．The inversion integral106

12．Adjacent strips of convergence for the same image109

13．Operational relations having a line of convergence113

14．Symmetry between the integrals of definition and inversion115

15．Uniqueness of the operational relations117

ⅦASYMPTOTIC RELATIONS AND OPERATIONAL TRANSPOSITION OF SERIES121

1．Introduction121

2．Abel and Tauber theorems122

3．Abel theorems124

4．Real Tauber theorems128

5．Complex Tauber theorems130

6．Operational equalities133

7．Asymptotic series134

8．Operational transposition of power series in p-1 and in t136

9．Transposition of series with ascending powers of p139

10．Expansion in rational fractions(Heaviside＇s expansion theorem Ⅱ)142

11．Transposition of other series147

12．A real inversion formula148

ⅧLINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS152

1．Introduction152

2．Inhomogeneous equations with the unit function at the right152

3．Inhomogeneous equations with arbitrary right-hand member155

4．Differential equations with boundary conditions157

5．Admittances and impedances161

6．Transient phenomena167

7．Time impedances170

8．Series for small values of t．Heaviside＇s first expansion theorem171

9．Classification of admittances173

ⅨSIMULTANEOUS LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS；ELECTRIC-CIRCUIT THEORY175

1．Introduction175

2．Equations of electric-circuit theory175

3．Transposition of the circuit equations；mutual impedances and admittances177

4．Time admittances of electric circuits180

5．Applications of the general circuit theory182

6．Circuit equations with initial conditions184

7．Ladder networks；filters185

8．Lattice networks194

ⅩLINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS200

1．Introduction200

2．Laguerre polynomials201

3．Hermite polynomials204

4．Bessel functions(operational relations)207

5．Bessel functions(applications)214

6．Legendre functions(operational relations)220

7．Legendre functions(applications)224

8．Hypergeometric functions225

ⅪOPERATIONAL RULES OF MORE COMPLICATED CHARACTER232

1．Introduction232

2．Rules obtained when p is replaced by a function of p232

3．Rules obtained when t is replaced by a function of t237

4．The exponential transformation：t→et238

5．Exponential operational relations for power series，in particular hypergeometric series243

6．Rules concerning series expansions249

7．Equalities in connexion with a single operational relation253

8．Equalities in connexion with a pair of simultaneous relations(exchange identity)254

ⅫSTEP FUNCTIONS AND OTHER DISCONTINUOUS FUNCTIONS257

1．Introduction257

2．Operational relations involving step functions jumping at integral values of t258

3．The saw-tooth function260

4．Arithmetic functions in connexion with θ-functions263

5．Arithmetic functions in connexion with Dirichlet series266

6．Step functions of argument equal to the summation variable in a series272

7．Contragrade series274

ⅩⅢDIFFERENCE EQUATIONS277

1．Introduction277

2．Difference equation for the ＇sum＇278

3．General linear difference equations with constant coefficients281

4．Linear difference equations with variable coefficients285

5．Connexion between differential and difference equations287

6．Operational construction of difference equations289

ⅩⅣINTEGRAL EQUATIONS292

1．Introduction292

2．The integral equation for the moving average293

3．Integral equations of the first kind with difference kernel300

4．Integral equations of the first kind with kernel reducible to a difference kernel305

5．Integral equations of the second kind with a difference kernel307

6．Homogeneous integral equations310

7．The operational construction of integral equations312

8．Note on the operational interpretation of the Wiener-Hopf technique313

ⅩⅤPARTIAL DIFFERENTIAL EQUATIONS IN THE OPERATIONAL CALCULUS OF ONE VARIABLE314

1．Introduction314

2．Homogeneous linear partial differential equations(general solutions)317

3．Homogeneous partial differential equations with boundary conditions322

4．Quasi-stationary theory of electric cables325

5．Inhomogeneous partial differential equations330

ⅩⅥSIMULTANEOUS OPERATIONAL CALCULUS334

1．Introduction334

2．General theory334

3．Second-order differential equations of the hyperbolic type with constant coefficients and two variables344

4．Hyperbolic differential equations of the second order in more than two independent variables352

5．Elliptic differential equations355

6．Simultaneous partial differential equations361

7．Partial difference equations365

ⅩⅦ＇GRAMMAR＇373

ⅩⅧ＇DICTIONARY＇383

LIST OF AUTHORS QUOTED411

GENERAL INDEX413