SPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS

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1 SPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS Second Edition LARRY C. ANDREWS OXFORD UNIVERSITY PRESS OXFORD TOKYO MELBOURNE SPIE OPTICAL ENGINEERING PRESS A Publication of SPIE The International Society for Optical Engineering Bellingham, Washington USA 1998

2 Preface to the Second Edition Preface to the First Edition Notation for Special Functions xi xiii xv Chapter 1. Infinite Series, Improper Integrals, and Infinite Products Introduction Infinite Series of Constants The Geometrie Series Summary of Convergence Tests Operations with Series Factorials and Binomial Coefficients Infinite Series of Functions Properties of Uniformly Convergent Series Power Series Sums and Products of Power Series Fourier Trigonometrie Series Cosine and Sine Series Improper Integrals Types of Improper Integrals Convergence Tests Pointwise and Uniform Convergence Asymptotic Formulas Small Arguments Large Arguments Infinite Products Associated Infinite Series Products of Functions 57 Chapter 2. The Gamma Function and Related Functions Introduction Gamma Function Integral Representations Legendre Duplication Formula Weierstrass'Infinite Product Applications Miscellaneous Problems Fractional-Order Derivatives Beta Function 82

3 vi 2.5 Incomplete Gamma Function Asymptotic Series Digamma and Polygamma Functions Integral Representations Asymptotic Series Polygamma Functions Riemann Zeta Function 102 Chapter 3. Other Functions Defined by Integrals Introduction Error Function and Related Functions Asymptotic Series Fresnel Integrals Applications Probability and Statistics Heat Conduction in Solids Vibrating Beams Exponential Integral and Related Functions Logarithmic Integral Sine and Cosine Integrals Elliptic Integrals Limiting Values and Series Representations The Pendulum Problem 135 Chapter 4. Legendre Polynomials and Related Functions Introduction Legendre Polynomials The Generating Function Special Values and Recurrence Formulas Legendre's Differential Equation Other Representations of the Legendre Polynomials Rodrigues'Formula Laplace Integral Formula SomeBoundson/^fX) Legendre Series Orthogonality of the Polynomials Finite Legendre Series Infinite Legendre Series Convergence of the Series Piecewise Continuous and Piecewise Smooth Functions Pointwise Convergence Legendre Functions of the Second Kind Basic Properties Associated Legendre Functions Basic Properties of P (x) Applications Electric Potential due to a Sphere Steady-State Temperatures in a Sphere 197

4 vii Chapter 5. Other Orthogonal Polynomials Introduction Hermite Polynomials Recurrence Formulas Hermite Series Simple Harmonie Oscillator Laguerre Polynomials Recurrence Formulas Laguerre Series Associated Laguerre Polynomials The Hydrogen Atom Generalized Polynomial Sets Gegenbauer Polynomials Chebyshev Polynomials Jacobi Polynomials 231 Chapter 6. Bessel Functions Introduction Bessel Functions of the First Kind The Generating Function Bessel Functions of the Nonintegral Order Recurrence Formulas Bessel's Differential Equation Integral Representations Bessel's Problem Geometrie Problems Integrals of Bessel Functions Indefinite Integrals Definite Integrals Series Involving Bessel Functions Addition Formulas Orthogonality of Bessel Functions Fourier-Bessel Series Bessel Functions of the Second Kind Series Expansion for Y n (x) Asymptotic Formulas for Small Arguments Recurrence Formulas Differential Equations Related to Bessel's Equation The Oscillating Chain 282 Chapter 7. Bessel Functions of Other Kinds Introduction Modified Bessel Functions Modified Bessel Functions of the Second Kind Recurrence Formulas Generating Function and Addition Theorems Integral Relations Integral Representations Integrals of Modified Bessel Functions Spherical Bessel Functions Recurrence Formulas Modified Spherical Bessel Functions 305

5 viii 7.5 Other Bessel Functions Hankel Functions Struve Functions Kelvin's Functions Airy Functions Asymptotic Formulas Small Arguments Large Arguments 317 Chapter 8. Applications Involving Bessel Functions Introduction Problems in Mechanics The Lengthening Pendulum Bückling of a Long Column Statistical Communication Theory Narrowband Noise and Envelope Detection Non-Rayleigh Radar Sea Clutter Heat Conduction and Vibration Phenomena Radial Symmetrie Problems Involving Circles Radial Symmetrie Problems Involving Cylinders The Helmholtz Equation Step-Index Optical Fibers 351 Chapter 9. The Hypergeometric Function Introduction The Pochhammer Symbol The Function F(a,b;c;x) Elementary Properties Integral Representation The Hypergeometric Equation Relation to Other Functions Legendre Functions Summing Series and Evaluating Integrals Action-Angle Variables 380 Chapter 10. The Confluent Hypergeometric Functions Introduction The Functions M(a;c;x) and U(a;c;x) Elementary Properties of M(a;c;x) Confluent Hypergeometric Equation and U(a;c;x) Asymptotic Formulas Relation to Other Functions Hermite Functions Laguerre Functions Whittaker Functions 403 Chapter 11. Generalized Hypergeometric Functions Introduction The Set of Functions pf q Hypergeometric-Type Series 413

6 ix 11.3 Other Generalizations The Meijer G Function The MacRobert E Function 425 Chapter 12. Applications Involving Hypergeometric-iype Functions Introduction Statistical Communication Theory Nonlinear Devices Fluid Mechanics Unsteady Hydrodynamic Flow Past an Infinite Plate Transonic Flow and the Euler-Tricomi Equation Random Fields Structure Function of Temperature 445 Bibliography 451 Appendix: A List of Special Function Formulas 453 Selected Answers to Exercises 469 Index 473

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