Special Functions and Their Applications by Lebedev, N.N. -- Read -- Imperial Library of Trantor

Index

Front Cover Title Page Copyright Author’s Preface Translator’s Preface Table of Contents 1 THE GAMMA FUNCTION

1.1. Definition of the Gamma Function 1.2. Some Relations Satisfied by the Gamma Function 1.3. The Logarithmic Derivative of the Gamma Function 1.4. Asymptotic Representation of the Gamma Function for Large |z| 1.5. Definite Integrals Related to the Gamma Function Problems

2 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS

2.1. The Probability Integral and Its Basic Properties 2.2. Asymptotic Representation of the Probability Integral for Large |z| 2.3. The Probability Integral of Imaginary Argument. The Function F(z) 2.4. The Probability Integral of Argument x. The Fresnel Integrals 2.5. Application to Probability Theory 2.6. Application to the Theory of Heat Conduction. Cooling of the Surface of a Heated Object 2.7. Application to the Theory of Vibrations. Transverse Vibrations of an Infinite Rod under the Action of a Suddenly Applied Concentrated Force Problems

3 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS

3.1. The Exponential Integral and its Basic Properties 3.2. Asymptotic Representation of the Exponential Integral for Large |z| 3.3. The Exponential Integral of Imaginary Argument. The Sine and Cosine Integrals 3.4. The Logarithmic Integral 3.5. Application to Electromagnetic Theory, Radiation of a Linear Half-Wave Oscillator Problems

4 ORTHOGONAL POLYNOMIALS

4.1. Introductory Remarks 4.2. Definition and Generating Function of the Legendre Polynomials 4.3. Recurrence Relations and Differential Equation for the Legendre Polynomials 4.4. Integral Representations of the Legendre Polynomials 4.5. Orthogonality of the Legendre Polynomials 4.6. Asymptotic Representation of the Legendre Polynomials for Large n 4.7. Expansion of Functions in Series of Legendre Polynomials 4.8. Examples of Expansions in Series of Legendre Polynomials 4.9. Definition and Generating Function of the Hermite Polynomials 4.10. Recurrence Relations and Differential Equation for the Hermite Polynomials 4.11. Integral Representations of the Hermite Polynomials 4.12. Integral Equations Satisfied by the Hermite Polynomials 4.13. Orthogonality of the Hermite Polynomials 4.14. Asymptotic Representation of the Hermite Polynomials for Large n 4.15. Expansion of Functions in Series of Hermite Polynomials 4.16. Examples of Expansions in Series of Hermite Polynomials 4.17. Definition and Generating Function of the Laguerre Polynomials 4.18. Recurrence Relations and Differential Equation for the Laguerre Polynomials 4.19. An Integral Representation of the Laguerre Polynomials. Relation between the Laguerre and Hermite Polynomials 4.20. An Integral Equation Satisfied by the Laguerre Polynomials 4.21. Orthogonality of the Laguerre Polynomials 4.22. Asymptotic Representation of the Laguerre Polynomials for Large n 4.23. Expansion of Functions in Series of Laguerre Polynomials 4.24. Examples of Expansions in Series of Laguerre Polynomials 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 Problems

5 CYLINDER FUNCTIONS: THEORY

5.1. Introductory Remarks 5.2. Bessel Functions of Nonnegative Integral Order 5.3. Bessel Functions of Arbitrary Order 5.4. General Cylinder Functions. Bessel Functions of the Second Kind 5.5. Series Expansion of the Function 5.6. Bessel Functions of the Third Kind 5.7. Bessel Functions of Imaginary Argument 5.8. Cylinder Functions of Half-Integral Order 5.9. Wronskians of Pairs of Solutions of Bessel’s Equation 5.10. Integral Representations of the Cylinder Functions 5.11. Asymptotic Representations of the Cylinder Functions for Large |z| 5.12. Addition Theorems for the Cylinder Functions, 124.Zeros of the Cylinder Functions 5.13. Expansions in Series and Integrals Involving Cylinder Functions 5.14. Definite Integrals Involving Cylinder Functions 5.15. Cylinder Functions of Nonnegative Argument and Order 5.16. Airy Functions Problems

6 CYLINDER FUNCTIONS: APPLICATIONS

6.1. Introductory Remarks 6.2. Separation of Variables in Cylindrical Coordinates 6.3. The Boundary Value Problems of Potential Theory. The Dirichlet Problem for a Cylinder 6.4. The Dirichlet Problem for a Domain Bounded by Two Parallel Planes 6.5. The Dirichlet Problem for a Wedge 6.6. The Field of a Point Charge near the Edge of a Conducting Sheet 6.7. Cooling of a Heated Cylinder 6.8. Diffraction by a Cylinder Problems

7 SPHERICAL HARMONICS: THEORY

7.1. Introductory Remarks 7.2. The Hypergeometric Equation and Its Series Solution 7.3. Legendre Functions 7.4. Integral Representations of the Legendre Functions 7.5. Some Relations Satisfied by the Legendre Functions 7.6. Series Representations of the Legendre Functions 7.7. Wronskians of Pairs of Solutions of Legend-re’s Equation 7.8. Recurrence Relations for the Legendre Functions 7.9. Legendre Functions of Nonnegative Integral Degree and Their Relation to Legendre Polynomials 7.10. Legendre Functions of Half-Integral Degree 7.11. Asymptotic Representations of the Legendre Functions for Large |v| 7.12. Associated Legendre Functions Problems

8 SPHERICAL HARMONICS: APPLICATIONS

8.1. Introductory Remarks 8.2. Solution of Laplace’s Equation in Spherical Coordinates 8.3. The Dirichlet Problem for a Sphere 8.4. The Field of a Point Charge Inside a Hollow Conducting Sphere 8.5. The Dirichlet Problem for a Cone 8.6. Solution of Laplace’s Equation in Spheroidal Coordinates 8.7. The Dirichlet Problem for a Spheroid 8.8. The Gravitational Attraction of a Homogeneous Solid Spheroid 8.9. The Dirichlet Problem for a Hyperboloid of Revolution 8.10. Solution of Laplace’s Equation in Toroidal Coordinates 8.11. The Dirichlet Problem for a Torus 8.12. The Dirichlet Problem for a Domain Bounded by Two Intersecting Spheres 8.13. Solution of Laplace’s Equation in Bipolar Coordinates 8.14. Solution of Helmholtz’s Equation in Spherical Coordinates Problems

9 HYPERGEOMETRIC FUNCTIONS

10 PARABOLIC CYLINDER FUNCTIONS

10.1. Separation of Variables in Laplace’s Equation in Parabolic Coordinates 10.2. Hermite Functions 10.3. Some Relations Satisfied by the Hermite Functions 10.4. Recurrence Relations for the Hermite Functions 10.5. Integral Representations of the Hermite Functions 10.6. Asymptotic Representations of the Hermite Functions for Large |z| 10.7. The Dirichlet Problem for a Parabolic Cylinder 10.8. Application to Quantum Mechanics Problems

Bibliography Index

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