Special functions of mathematics for engineers:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
McGraw-Hill
1992
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| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 1. Aufl. u.d.T.: Andrews, Larry C.: Special functions for engineers and applied mathematicians |
| Physical Description: | XIX, 479 S. graph. Darst. |
| ISBN: | 0070018480 |
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| adam_text | Titel: Special functions of mathematics for engineers
Autor: Andrews, Larry C
Jahr: 1992
Contents Preface to the Second Edition xiii Preface to the First Edition xv Notation for Special Functions xvii Chapter 1. Infinite Series, Improper Integrals, and Infinite Products 1 1.1 Introduction 1 1.2 Infinite Series of Constants 2 1.2.1 The Geometric Series 4 1.2.2 Summary of Convergence Tests 6 1.2.3 Operations with Series 11 1.2.4 Factorials and Binomial Coefficients 15 1.3 Infinite Series of Functions 21 1.3.1 Properties of Uniformly Convergent Series 23 1.3.2 Power Series 25 1.3.3 Sums and Products of Power Series 29 1.4 Fourier Trigonometric Series 33 1.4.1 Cosine and Sine Series 36 1.5 Improper Integrals 39 1.5.1 Types of Improper Integrals 39 1.5.2 Convergence Tests 42 1.5.3 Pointwise and Uniform Convergence 43 1.6 Asymptotic Formulas 47 1.6.1 Small Arguments 48 1.6.2 Large Arguments 50 1.7 Infinite Products 55 1.7.1 Associated Infinite Series 56 1.7.2 Products of Functions 57 Chapter 2. The Gamma Function and Related Functions 61 2.1 Introduction 61 2.2 Gamma Function 62 2.2.1 Integral Representations 64 2.2.2 Legendre Duplication Formula 70 2.2.3 Weierstrass’ Infinite Product 71 vil
vili Contents 2.3 Applications 77 2.3.1 Miscellaneous Problems 77 2.3.2 Fractional-Order Derivatives 79 2.4 Beta Function 82 2.5 Incomplete Gamma Function 87 2.5.1 Asymptotic Series 88 2.6 Digamma and Polygamma Functions 90 2.6.1 Integral Representations 93 2.6.2 Asymptotic Series 95 2.6.3 Polygamma Functions 100 2.6.4 Riemann Zeta Function 102 Chapter 3. Other Functions Defined by Integrals 109 3.1 Introduction 109 3.2 Error Function and Related Functions 110 3.2.1 Asymptotic Series 112 3.2.2 Fresnel Integrals 113 3.3 Applications 118 3.3.1 Probability and Statistics 118 3.3.2 Heat Conduction in Solids 119 3.3.3 Vibrating Beams 122 3.4 Exponential Integral and Related Functions 126 3.4.1 Logarithmic Integral 128 3.4.2 Sine and Cosine Integrals 129 3.5 Elliptic Integrals 133 3.5.1 Limiting Values and Series Representations 134 3.5.2 The Pendulum Problem 135 Chapter 4. Legendre Polynomials and Related Functions 141 4.1 Introduction 141 4.2 Legendre Polynomials 142 4.2.1 The Generating Function _ 142 4.2.2 Special Values and Recurrence Formulas 146 4.2.3 Legendre’s Differential Equation 151 4.3 Other Representations of the Legendre Polynomials 157 4.3.1 Rodrigues’ Formula 157 4.3.2 Laplace Integral Formula 158 4.3.3 Some Bounds on P„(x) 159 4.4 Legendre Series 162 4.4.1 Orthogonality of the Polynomials 162 4.4.2 Finite Legendre Series 165 4.4.3 Infinite Legendre Series 167 4.5 Convergence of the Series 173 4.5.1 Piecewise Continuous and Piecewise Smooth Functions 174 4.5.2 Pointwise Convergence 175 4.6 Legendre Functions of the Second Kind 181 4.6.1 Basic Properties 184 4.7 Associated Legendre Functions 186 4.7.1 Basic Properties of P„(x) 189
Contents fx 4.8 Applications 192 4.8.1 Electric Potential due to a Sphere 193 4.8.2 Steady-State Temperatures In a Sphere 197 Chapter 5. Other Orthogonal Polynomials 203 5.1 Introduction 203 5.2 Hermite Polynomials 204 5.2.1 Recurrence Formulas 206 5.2.2 Hermite Series 207 5.2.3 Simple Harmonic Oscillator 209 5.3 Laguerre Polynomials 214 5.3.1 Recurrence Formulas 215 5.3.2 Laguerre Series 217 5.3.3 Associated Laguerre Polynomials 218 5.3.4 The Hydrogen Atom 221 5.4 Generalized Polynomial Sets 226 5.4.1 Gegenbauer Polynomials 226 5.4.2 Chebyshev Polynomials 228 5.4.3 Jacobi Polynomials 231 Chapter 6. Bessel Functions 237 6.1 Introduction 237 6.2 Bessel Functions of the First Kind 238 6.2.1 The Generating Function 238 6.2.2 Bessel Functions of Nonintegral Order 240 6.2.3 Recurrence Formulas 242 6.2.4 Bessel’s Differential Equation 243 6.3 Integral Representations 248 6.3.1 Bessel’s Problem 250 6.3.2 Geometric Problems 253 6.4 Integrals of Bessel Functions 256 6.4.1 Indefinite Integrals 256 6.4.2 Definite Integrals 258 6.5 Series Involving Bessel Functions 265 6.5.1 Addition Formulas 265 6.5.2 Orthogonality of Bessel Functions 267 6.5.3 Fourler-Bessel Series 269 6.6 Bessel Functions of the Second Kind 273 6.6.1 Series Expansion for Y n (x) 274 6.6.2 Asymptotic Formulas for Small Arguments 277 6.6.3 Recurrence Formulas 278 6.7 Differential Equations Related to Bessel’s Equation 280 6.7.1 The Oscillating Chain 282 Chapter 7. Bessel Functions of Other Kinds 287 7.1 Introduction 287 7.2 Modified Bessel Functions 287 7.2.1 Modified Bessel Functions of the Second Kind 290
x Contents 7.2.2 Recurrence Formulas 291 7.2.3 Generating Function and Addition Theorems 292 7.3 Integral Relations 298 7.3.1 Integral Representations 298 7.3.2 Integrals of Modified Bessel Functions 299 7.4 Spherical Bessel Functions 302 7.4.1 Recurrence Formulas 305 7.4.2 Modified Spherical Bessel Functions 305 7.5 Other Bessel Functions 308 7.5.1 Hankel Functions 308 7.5.2 Struve Functions 309 7.5.3 Kelvin s Functions 311 7.5.4 Airy Functions 312 7.6 Asymptotic Formulas 316 7.6.1 Small Arguments 316 7.6.2 Large Arguments 317 Chapter 8. Applications Involving Bessel Functions 323 8.1 Introduction 323 8.2 Problems In Mechanics 323 8.2.1 The Lengthening Pendulum 323 8.2.2 Buckling of a Long Column 327 8.3 Statistical Communication Theory 332 8.3.1 Narrowband Noise and Envelope Detection 333 8.3.2 Non-Rayleigh Radar Sea Clutter 336 8.4 Heat Conduction and Vibration Phenomena 339 8.4.1 Radial Symmetric Problems Involving Circles 340 8.4.2 Radial Symmetric Problems Involving Cylinders 343 8.4.3 The Helmholtz Equation 345 8.5 Step-Index Optical Fibers 351 Chapter 9. The Hypergeometric Function 357 9.1 Introduction 357 9.2 The Pochhammer Symbol 358 9.3 The Function F(a, b;c;x) 361 9.3.1 Elementary Properties 362 9.3.2 Integral Representation 364 9.3.3 The Hypergeometric Equation 365 9.4 Relation to Other Functions 370 9.4.1 Legendre Functions 373 9.5 Summing Series and Evaluating Integrals 377 9.5.1 Action-Angle Variables 380 Chapter 10. The Confluent Hypergeometric Functions 385 10.1 Introduction 385 10.2 The Functions M(a; c; x) and U(a; c; x) 386
Contents xi 10.2.1 Elementary Properties of Af(a; c; x) 386 10.2.2 Confluent Hypergeometric Equation and U(a; c; x) 388 10.2.3 Asymptotic Formulas 390 10.3 Relation to Other Functions 395 10.3.1 Hermite Functions 397 10.3.2 Laguerre Functions 399 10.4 Whittaker Functions 403 Chapter 11. Generalized Hypergeometric Functions 411 11.1 Introduction 411 11.2 The Set of Functions p F q 412 11.2.1 Hypergeometric-Type Series 413 11.3 Other Generalizations 419 11.3.1 The Meijer G Function 419 11.3.2 The MacRobert E Function 425 Chapter 12. Applications Involving Hypergeometric-Type Functions 429 12.1 Introduction 429 12.2 Statistical Communication Theory 429 12.2.1 Nonlinear Devices 431 12.3 Fluid Mechanics 437 12.3.1 Unsteady Hydrodynamic Flow Past an Infinite Plate 437 12.3.2 Transonic Flow and the Euler-Tricomi Equation 440 12.4 Random Fields 444 12.4.1 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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| discipline | Mathematik |
| edition | 2. ed. |
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| indexdate | 2024-07-09T16:41:34Z |
| institution | BVB |
| isbn | 0070018480 |
| language | English |
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| spelling | Andrews, Larry C. Verfasser aut Special functions of mathematics for engineers Larry C. Andrews 2. ed. New York McGraw-Hill 1992 XIX, 479 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier 1. Aufl. u.d.T.: Andrews, Larry C.: Special functions for engineers and applied mathematicians Fonctions spéciales ram Mathématiques de l'ingénieur ram Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Spezielle Funktion (DE-588)4182213-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003907474&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Andrews, Larry C. Special functions of mathematics for engineers Fonctions spéciales ram Mathématiques de l'ingénieur ram Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd |
| subject_GND | (DE-588)4182213-4 |
| title | Special functions of mathematics for engineers |
| title_auth | Special functions of mathematics for engineers |
| title_exact_search | Special functions of mathematics for engineers |
| title_full | Special functions of mathematics for engineers Larry C. Andrews |
| title_fullStr | Special functions of mathematics for engineers Larry C. Andrews |
| title_full_unstemmed | Special functions of mathematics for engineers Larry C. Andrews |
| title_short | Special functions of mathematics for engineers |
| title_sort | special functions of mathematics for engineers |
| topic | Fonctions spéciales ram Mathématiques de l'ingénieur ram Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd |
| topic_facet | Fonctions spéciales Mathématiques de l'ingénieur Functions, Special Spezielle Funktion |
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