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E-raamat: Special Functions and their Application

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  • Formaat: 122 pages
  • Ilmumisaeg: 01-Sep-2022
  • Kirjastus: River Publishers
  • ISBN-13: 9781000792270
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Special Functions and their ApplicationSuurem pilt
  • Formaat: 122 pages
  • Ilmumisaeg: 01-Sep-2022
  • Kirjastus: River Publishers
  • ISBN-13: 9781000792270
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This short text gives clear descriptions and explanations of the Gamma function, the Probability Integral and its related functions, Spherical Harmonics Theory, The Bessel function, Hermite polynomials and Laguerre polynomials.

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 Integral and its related functions, Spherical Harmonics Theory, The Bessel function, Hermite polynomials and Laguerre polynomials. Each chapter finishes with a description of how the function is most commonly applied and a set of examples for the student to work through.
Preface ix
List of Tables
xi
List of Abbreviations
xiii
1 The Gamma Function
1(14)
1.1 Definition of Gamma Function
1(2)
1.2 Gamma Function and Some Relations
3(3)
1.3 The Logarithmic Derivative of the Gamma Function
6(4)
1.4 Asymptotic Representation of the Gamma Function for Large |z|
10(1)
1.5 Definite Integrals Related to the Gamma Function
11(1)
1.6 Exercises
12(3)
2 The Probability Integral and Related Functions
15(16)
2.1 The Probability Integral and its Basic Properties
15(2)
2.2 Asymptotic Representation of Probability Integral for Large |z|
17(1)
2.3 The Probability Integral of Imaginary Argument
18(2)
2.4 The Probability Fresnel Integrals
20(3)
2.5 Application to Probability Theory
23(1)
2.6 Application to the Theory of Heat Conduction
24(2)
2.7 Application to the Theory of Vibrations
26(2)
2.8 Exercises
28(3)
3 Spherical Harmonics Theory
31(18)
3.1 Introduction
31(1)
3.2 The Hypergeometric Equation and its Series Solution
32(3)
3.3 Legendre Functions
35(2)
3.4 Integral Representations of the Legendre Functions
37(2)
3.5 Some Relations Satisfied by the Legendre Functions
39(1)
3.6 Workskian of Pairs of Solutions of Legendre's Equation
40(2)
3.7 Recurrence Relations for the Legendre Functions
42(2)
3.8 Associated Legendre Functions
44(2)
3.9 Exercises
46(3)
4 Bessel Function
49(16)
4.1 Bessel Functions
49(5)
4.2 Generating Function
54(3)
4.3 Recurrence Relations
57(2)
4.4 Orthonormality
59(1)
4.5 Application to the Optical Fiber
60(2)
4.6 Exercises
62(3)
5 Hermite Polynomials
65(16)
5.1 Hermite Functions
65(4)
5.2 Generating Function
69(1)
5.3 Recurrence Relations
70(3)
5.4 Rodrigues Formula
73(1)
5.5 Orthogonality and Normalilty
74(2)
5.6 Application to the Simple Harmonic Oscillator
76(2)
5.7 Exercises
78(3)
6 Laguerre Polynomials
81(24)
6.1 Laguerre Functions
81(4)
6.2 Generating Function
85(2)
6.3 Recurrence Relations
87(4)
6.4 Rodrigues Formula
91(1)
6.5 Orthonormality
92(2)
6.6 Application to the Hydrogen Atom
94(4)
6.7 Associated Laguerre Polynomials
98(4)
6.7.1 Properties of Associated Laguerre Polynomials
102(1)
6.8 Exercises
102(3)
Bibliography 105(2)
Index 107(2)
About the Authors 109
Bipin Singh Koranga, Sanjay Kumar Padaliya, Vivek Kumar Nautiyal
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