Special Functions Of Mathematics For Engineers

Modern Engineering And Physical Science Applications Demand A Thorough Knowledge Of Applied Mathematics, Particularly Special Functions. These Typically Arise In Applications Such As Communication Systems, Electro-optics, Nonlinear Wave Propagation, Electromagnetic Theory, Electric Circuit Theory, And Quantum Mechanics. This Text Systematically Introduces Special Functions And Explores Their Properties And Applications In Engineering And Science. Chapter 1. Infinite Series, Improper Integrals, And Infinite Products -- Introduction -- Infinite Series Of Constants -- The Geometric 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 Trigonometric 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. 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 -- Incomplete Gamma Function -- Asymptotic Series -- Digamma And Polygamma Functions -- Integral Representations -- Asymptotic Series -- Polygamma Functions -- Riemann Zeta Function. 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. 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 -- Some Bounds On Pn(x) -- 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 Pmn(x) -- Applications -- Electric Potential Due To A Sphere -- Steady-state Temperatures In A Sphere. Chapter 5. Other Orthogonal Polynomials -- Introduction -- Hermite Polynomials -- Recurrence Formulas -- Hermite Series -- Simple Harmonic Oscillator -- Laguerre Polynomials -- Recurrence Formulas -- Laguerre Series -- Associated Laguerre Polynomials -- The Hydrogen Atom -- Generalized Polynomial Sets -- Gegenbauer Polynomials -- Chebyshev Polynomials -- Jacobi Polynomials. 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 -- Geometric 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 Yn(x) -- Asymptotic Formulas For Small Arguments -- Recurrence Formulas -- Differential Equations Related To Bessel's Equation -- The Oscillating Chain. 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 -- Other Bessel Functions -- Hankel Functions -- Struve Functions -- Kelvin's Functions -- Airy Functions -- Asymptotic Formulas -- Small Arguments -- Large Arguments. Chapter 8. Applications Involving Bessel Functions -- Introduction -- Problems In Mechanics -- The Lengthening Pendulum -- Buckling Of A Long Column -- Statistical Communication Theory -- Narrowband Noise And Envelope Detection -- Non-rayleigh Radar Sea Clutter -- Heat Conduction And Vibration Phenomena -- Radial Symmetric Problems Involving Circles -- Radial Symmetric Problems Involving Cylinders -- The Helmholtz Equation -- Step-index Optical Fibers -- 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. 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 -- Chapter 11. Generalized Hypergeometric Functions -- Introduction -- The Set Of Functions Pfq -- Hypergeometric-type Series -- Other Generalizations -- The Meijer G Function -- The Macrobert E Function -- Chapter 12. Applications Involving Hypergeometric-type 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 -- Bibliography -- Appendix: A List Of Special Function Formulas -- Selected Answers To Exercises -- Index. Larry C. Andrews. Originally Published: 2nd Ed. New York : Mcgraw-hill, C1992. Includes Bibliographical References (p. 451) And Index.