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.
Author(s): Bipin Singh Koranga, Sanjay Kumar Padaliya, Vivek Kumar Nautiyal
Series: River Publishers Series in Mathematical and Engineering Sciences
Publisher: River Publishers
Year: 2021
Language: English
Pages: 122
City: Gistrup
Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
List of Tables
1: The Gamma Function
1.1 Definition of Gamma Function
1.2 Gamma Function and Some Relations
1.3 The Logarithmic Derivative of the Gamma Function
1.4 Asymptotic Representation of the Gamma Function for Large |z|
1.5 Definite Integrals Related to the Gamma Function
1.6 Exercises
2: The Probability Integral and Related Functions
2.1 The Probability Integral and its Basic Properties
2.2 Asymptotic Representation of Probability Integral for Large |z|
2.3 The Probability Integral of Imaginary Argument
2.4 The Probability Fresnel Integrals
2.5 Application to Probability Theory
2.6 Application to the Theory of Heat Conduction
2.7 Application to the Theory of Vibrations
2.8 Exercises
3: Spherical Harmonics Theory
3.1 Introduction
3.2 The Hypergeometric Equation and its Series Solution
3.3 Legendre Functions
3.4 Integral Representations of the Legendre Functions
3.5 Some Relations Satisfied by the Legendre Functions
3.6 Workskian of Pairs of Solutions of Legendre’s Equation
3.7 Recurrence Relations for the Legendre Functions
3.8 Associated Legendre Functions
3.9 Exercises
4: Bessel Function
4.1 Bessel Functions
4.2 Generating Function
4.3 Recurrence Relations
4.4 Orthonormality
4.5 Application to the Optical Fiber
4.6 Exercises
5: Hermite Polynomials
5.1 Hermite Functions
5.2 Generating Function
5.3 Recurrence Relations
5.4 Rodrigues Formula
5.5 Orthogonality and Normalilty
5.6 Application to the Simple Harmonic Oscillator
5.7 Exercises
6: Laguerre Polynomials
6.1 Laguerre Functions
6.2 Generating Function
6.3 Recurrence Relations
6.4 Rodrigues Formula
6.5 Orthonormality
6.6 Application to the Hydrogen Atom
6.7 Associated Laguerre Polynomials
6.7.1 Properties of Associated Laguerre Polynomials
6.8 Exercises
Bibliography
Index
About the Authors
