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Ⅰ GENERAL INTRODUCTION 1

1．History of the operational calculus 1

2．The operational calculus based on the Laplace transform 2

3．Survey of the subject-matter 5

Ⅱ THE FOURIER INTEGRAL AS BASIS OP THE OPERATIONAL CALCULUS 7

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(5b) 11

4．Rotation of the path of integration in the integral(11b) 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 22

1．Introduction 22

2．Images of polynomials；the Γ-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 33

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 tn 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 56

1．Introduction 56

2．The unit function 56

3．The δ-function as derivative of the unit function 59

4．History of the impulse function 62

5．The sifting integral as a Stieltjes integral 66

6．The image of the δ-function 68

7．Functions approximating the δ-function 70

8．Applications of the δ-function 74

9．Series of impulse functions 78

10．Derivatives of the δ-function 80

11．Note on the theory of distributions 84

Ⅵ QUESTIONS CONCERNING THE CONVERGENCE OF THE DEFINITION INTEGRAL 85

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 115

15．Uniqueness of the operational relations 117

Ⅶ ASYMPTOTIC RELATIONS AND OPERATIONAL TRANSPOSITION OF SERIES 121

1．Introduction 121

2．Abel and Tauber theorems 122

3．Abel theorems 124

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

Ⅷ LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 152

1．Introduction 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 175

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 200

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 232

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：t→et 238

5．Exponential operational relations for power series，in particular hypergeometric series 243

6．Rules concerning series expansions 249

7．Equalities in connexion with a single operational relation 253

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

Ⅻ STEP FUNCTIONS AND OTHER DISCONTINUOUS FUNCTIONS 257

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 θ-functions 263

5．Arithmetic 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

ⅩⅢ DIFFERENCE EQUATIONS 277

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 292

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 314

1．Introduction 314

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

3．Homogeneous partial differential equations with boundary conditions 322

4．Quasi-stationary theory of electric cables 325

5．Inhomogeneous partial differential equations 330

ⅩⅥ SIMULTANEOUS OPERATIONAL CALCULUS 334

1．Introduction 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

ⅩⅦ ＇GRAMMAR＇ 373

ⅩⅧ ＇DICTIONARY＇ 383

LIST OF AUTHORS QUOTED 411

GENERAL INDEX 413