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Special functions and their applications /
Special functions are mathematical functions that have established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. This short text gives clear descriptions and explanations of the Gamma function, the Probability Int...
| Clasificación: | Libro Electrónico |
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| Autores principales: | , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
[Place of publication not identified] :
River Publishers,
2021.
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| Colección: | River Publishers series in mathematical and engineering sciences.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface ix List of Tables xi List of Abbreviations xiii 1 The Gamma Function 1 1.1 Definition of Gamma Function 1 1.2 Gamma Function and Some Relations 3 1.3 The Logarithmic Derivative of the Gamma Function 6 1.4 Asymptotic Representation of the Gamma Function for Large z 10 1.5 Definite Integrals Related to the Gamma Function 11 1.6 Exercises 12 2 The Probability Integral and Related Functions 15 2.1 The Probability Integral and its Basic Properties 15 2.2 Asymptotic Representation of Probability Integral for Large z 17 2.3 The Probability Integral of Imaginary Argument 18 2.4 The Probability Fresnel Integrals 20 2.5 Application to Probability Theory 23 2.6 Application to the Theory of Heat Conduction 24 2.7 Application to the Theory of Vibrations 26 2.8 Exercises 28 3 Spherical Harmonics Theory 31 3.1 Introduction 31 3.2 The Hypergeometric Equation and its Series Solution 32 3.3 Legendre Functions 35 3.4 Integral Representations of the Legendre Functions 37 3.5 Some Relations Satisfied by the Legendre Functions 39 3.6 Workskian of Pairs of Solutions of Legendre's Equation 40 3.7 Recurrence Relations for the Legendre Functions 42 3.8 Associated Legendre Functions 44 3.9 Exercises 46 4 Bessel Function 49 4.1 Bessel Functions 49 4.2 Generating Function 54 4.3 Recurrence Relations 57 4.4 Orthonormality 59 4.5 Application to the Optical Fiber 60 4.6 Exercises 62 5 Hermite Polynomials 65 5.1 Hermite Functions 65 5.2 Generating Function 69 5.3 Recurrence Relations 70 5.4 Rodrigues Formula 73 5.5 Orthogonality and Normalilty 74 5.6 Application to the Simple Harmonic Oscillator 76 5.7 Exercises 78 6 Laguerre Polynomials 81 6.1 Laguerre Functions 81 6.2 Generating Function 85 6.3 Recurrence Relations 87 6.4 Rodrigues Formula 91 6.5 Orthonormality 92 6.6 Application to the Hydrogen Atom 94 6.7 Associated Laguerre Polynomials 98 6.7.1 Properties of Associated Laguerre Polynomials 102 6.8 Exercises 102 Bibliography 105 Index 107 About the Authors 109.