Special functions and their applications

Cover......Page 1
Title page......Page 3
Date-line......Page 4
Author's preface......Page 5
Translator's preface......Page 8
CONTENTS......Page 9
1.1. Definition of the Gamma Function......Page 15
1.2. Some Relations Satisfied by the Gamma Function......Page 17
1.3. The Logarithmic Derivative of the Gamma Function......Page 19
1.4. Asymptotic Representation of the Gamma Function for Large $|z|$......Page 22
1.5. Definite Integrals Related to the Gamma Function......Page 27
Problems......Page 28
2.1. The Probability Integral and Its Basic Properties......Page 30
2.2. Asymptotic Representation of the Probability Integral for Large $|z|$......Page 32
2.3. The Probability Integral of Imaginary Argument. The Function $F(z)$......Page 33
2.4. The Probability Integral of Argument $\sqrt{i}x$. The Fresnel Integrals......Page 35
2.5. Application to Probability Theory......Page 37
2.6. Application to the Theory of Heat Conduction. Cooling of the Surface of a Heated Object......Page 38
2.7. Application to the Theory of Vibrations. Transverse Vibrations of an Infinite Rod under the Action of a Suddenly Applied Concentrated Force......Page 40
Problems......Page 42
3.1. The Exponential Integral and its Basic Properties......Page 44
3.2. Asymptotic Representation of the Exponential Integral for Large $|z|$......Page 46
3.3. The Exponential Integral of Imaginary Argument. The Sine and Cosine Integrals......Page 47
3.4. The Logarithmic Integral......Page 51
3.5. Application to Electromagnetic Theory, Radiation of a Linear Half-Wave Oscillator......Page 53
Problems......Page 55
4.1. Introductory Remarks......Page 57
4.2. Definition and Generating Function of the Legendre Polynomials......Page 58
4.3. Recurrence Relations and Differential Equation for the Legendre Polynomials......Page 60
4.4. Integral Representations of the Legendre Polynomials......Page 62
4.5. Orthogonality of the Legendre Polynomials......Page 64
4.6. Asymptotic Representation of the Legendre Polynomials for Large $n$......Page 65
4.7. Expansion of Functions in Series of Legendre Polynomials......Page 67
4.8. Examples of Expansions in Series of Legendre Polynomials......Page 72
4.9. Definition and Generating Function of the Hermite Polynomials......Page 74
4.10. Recurrence Relations and Differential Equation for the Hermite Polynomials......Page 75
4.11. Integral Representations of the Hermite Polynomials......Page 77
4.12. Integral Equations Satisfied by the Hermite Polynomials......Page 78
4.13. Orthogonality of the Hermite Polynomials......Page 79
4.14. Asymptotic Representation of the Hermite Polynomials for Large $n$......Page 80
4.15. Expansion of Functions in Series of Hermite Polynomials......Page 82
4.16. Examples of Expansions in Series of Hermite Polynomials......Page 87
4.17. Definition and Generating Function of the Laguerre Polynomials......Page 90
4.18. Recurrence Relations and Differential Equation for the Laguerre Polynomials......Page 92
4.19. An Integral Representation of the Laguerre Polynomials. Relation between the Laguerre and Hermite Polynomials......Page 94
4.20. An Integral Equation Satisfied by the Laguerre Polynomials......Page 96
4.21. Orthogonality of the Laguerre Polynomials......Page 97
4.22. Asymptotic Representation of the Laguerre Polynomials for Large $n$......Page 99
4.24. Examples of Expansions in Series of Laguerre Polynomials......Page 102
4.25. Application to the Theory of Propagation of Electromagnetic Waves. Reflection from the End of a Long Transmission Line Terminated by a Lumped Inductance......Page 105
Problems......Page 107
5.1. Introductory Remarks......Page 112
5.2. Bessel Functions of Nonnegative Integral Order......Page 113
5.3. Bessel Functions of Arbitrary Order......Page 116
5.4. General Cylinder Functions. Bessel Functions of the Second Kind......Page 118
5.5. Series Expansion of the Function $Y_n(z)$......Page 120
5.6. Bessel Functions of the Third Kind......Page 121
5.7. Bessel Functions of Imaginary Argument......Page 122
5.8. Cylinder Functions of Half-Integral Order......Page 125
5.9. Wronskians of Pairs of Solutions of Bessel's Equation......Page 126
5.10. Integral Representations of the Cylinder Functions......Page 127
5.11. Asymptotic Representations of the Cylinder Functions for Large $|z|$......Page 134
5.12. Addition Theorems for the Cylinder Functions......Page 138
5.13. Zeros of the Cylinder Functions......Page 140
5.14. Expansions in Series and Integrals Involving Cylinder Functions......Page 142
5.15. Definite Integrals Involving Cylinder Functions......Page 145
5.16. Cylinder Functions of Nonnegative Argument and Order......Page 148
5.17. Airy Functions......Page 150
Problems......Page 153
6.2. Separation of Variables in Cylindrical Coordinates......Page 157
6.3. The Boundary Value Problems of Potential Theory. The Dirichlet Problem for a Cylinder......Page 160
6.4. The Dirichlet Problem for a Domain Bounded by Two Parallel Planes......Page 163
6.5. The Dirichlet Problem for a Wedge......Page 164
6.6. The Field of a Point Charge near the Edge of a Conducting Sheet......Page 167
6.7. Cooling of a Heated Cylinder......Page 169
6.8. Diffraction by a Cylinder......Page 170
Problems......Page 172
7.1. Introductory Remarks......Page 175
7.2. The Hypergeometric Equation and Its Series Solution......Page 176
7.3. Legendre Functions......Page 178
7.4. Integral Representations of the Legendre Functions......Page 185
7.5. Some Relations Satisfied by the Legendre Functions......Page 188
7.6. Series Representations of the Legendre Functions......Page 190
7.7. Wronskians of Pairs of Solutions of Legendre's Equation......Page 195
7.8. Recurrence Relations for the Legendre Functions......Page 197
7.9. Legendre Functions of Nonnegative Integral Degree and Their Relation to Legendre Polynomials......Page 198
7.10. Legendre Functions of Half-Integral Degree......Page 200
7.11. Asymptotic Representations of the Legendre Functions for Large $|\nu|$......Page 203
7.12. Associated Legendre Functions......Page 206
Problems......Page 213
8.1. Introductory Remarks......Page 218
8.2. Solution of Laplace's Equation in Spherical Coordinates......Page 219
8.3. The Dirichlet Problem for a Sphere......Page 220
8.4. The Field of a Point Charge Inside a Hollow Conducting Sphere......Page 222
8.5. The Dirichlet Problem for a Cone......Page 224
8.6. Solution of Laplace's Equation in Spheroidal Coordinates......Page 227
8.7. The Dirichlet Problem for a Spheroid......Page 229
8.8. The Gravitational Attraction of a Homogeneous Solid Spheroid......Page 232
8.9. The Dirichlet Problem for a Hyperboloid of Revolution......Page 234
8.10. Solution of Laplace's Equation in Toroidal Coordinates......Page 235
8.11. The Dirichlet Problem for a Torus......Page 238
8.12. The Dirichlet Problem for a Domain Bounded by Two Intersecting Spheres......Page 241
8.13. Solution of Laplace's Equation in Bipolar Coordinates......Page 244
8.14. Solution of Helmholtz's Equation in Spherical Coordinates......Page 248
Problems......Page 249
9.1. The Hyper geometric Series and Its Analytic Continuation......Page 252
9.2. Elementary Properties of the Hypergeometric Function......Page 255
9.3. Evaluation of $\lim\limits_{z\to 1-} F(\alpha,\beta;\gamma;z)$ for $\Re(\gamma - \alpha - \beta) > 0 $ ......Page 257
9.4. $F(\alpha,\beta;\gamma;z)$ as a Function of its Parameters......Page 259
9.5. Linear Transformations of the Hypergeometric Function......Page 260
9.6. Quadratic Transformations of the Hypergeometric Function......Page 264
9.7. Formulas for Analytic Continuation of $F(\alpha,\beta;\gamma;z)$ in Exceptional Cases......Page 270
9.8. Representation of Various Functions in Terms of the Hypergeometric Function......Page 272
9.9. The Confluent Hypergeometric Function......Page 274
9.10. The Differential Equation for the Confluent Hypergeometric Function and Its Solution. The Confluent Hypergeometric Function of the Second Kind......Page 276
9.11. Integral Representations of the Confluent Hypergeometric Functions......Page 280
9.12. Asymptotic Representations of the Confluent Hypergeometric Functions for Large $|z|$......Page 282
9.13. Representation of Various Functions in Terms of the Confluent Hypergeometric Functions......Page 285
9.14. Generalized Hypergeometric Functions......Page 289
Problems......Page 290
10.1. Separation of Variables in Laplace's Equation in Parabolic Coordinates......Page 295
10.2. Hermite Functions......Page 297
10.3. Some Relations Satisfied by the Hermite Functions......Page 301
10.4. Recurrence Relations for the Hermite Functions......Page 302
10.5. Integral Representations of the Hermite Functions......Page 304
10.6. Asymptotic Representations of the Hermite Functions for Large $|z|$......Page 305
10.7. The Dirichlet Problem for a Parabolic Cylinder......Page 307
Problems......Page 311
BIBLIOGRAPHY......Page 314
INDEX......Page 318