River Publishers Series in Mathematical and Engineering Sciences Series

Special Functions and Their Applications

January 2022
More details
  • Publisher
    River Publishers
  • Published
    24th January
  • ISBN 9788770226264
  • Language English
  • Pages 155 pp.
  • Size 6" x 9"
Lib E-Book

Library E-Books

We are signed up with aggregators who resell networkable e-book editions of our titles to academic libraries. These editions, priced at par with simultaneous hardcover editions of our titles, are not available direct from Stylus.

These aggregators offer a variety of plans to libraries, such as simultaneous access by multiple library patrons, and access to portions of titles at a fraction of list price under what is commonly referred to as a "patron-driven demand" model.

December 2021
More details
  • Publisher
    River Publishers
  • Published
    29th December 2021
  • ISBN 9788770226257
  • Language English
  • Pages 155 pp.
  • Size 6" x 9"

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.

1  The Gamma Function

1.1 Definition of Gamma function

1.2 Gamma Function 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 & 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

Bipin Singh Koranga, PhD

Bipin Singh Koranga is a Graduate from Kumaun University, Nainital. He has been with the Theoretical Physics Group, IIT Bombay since 2006 and received his Ph.D. in Physics (Neutrino Masses and Mixings) from the Indian Institute of Technology Bombay in 2007. He has been teaching basic courses in Physics and Mathematical Physics at the graduate level for the last 12 years. His research interests include the origin of the universe, Physics beyond the standard model, theoretical nuclear Physics, quantum mechanical neutrino oscillation and some topics related to astronomy. He has published over 42 scientific papers in various international journals. His present research interests include the neutrino mass models and related phenomenology.

Sanjay Kumar Padaliya, PhD

Sanjay Kumar Padaliya is presently Head, Department of Mathematics, S.G.R.R. (P.G) College, Dehradun. He received his Ph.D. in Mathematics (Fixed Point Theory) from Kumaun University, Nainital. He has been teaching basic courses in Mathematics at graduate and postgraduate level for the last 20 years. His present research interests include Fixed Point Theory and Fuzzy Analysis. He has published over 25 scientific papers in various international journals of repute and has presented his works at national and international conferences. His latest book, An Introduction to Tensor Analysis is published by River Publishers. Dr. Padaliya is also a life member of Indian Mathematical Society, Ramanujan Mathematical Society and International Academy of Physical Sciences.

Vivek Kumar Nautiyal, PhD

Vivek Kumar Nautiyal graduated from Lucknow University, Lucknow. He did his M.Tech in Applied Optics from Indian Institute of Technology Delhi in 2014. He has remained up-to-date with Nuclear and Particle physics since 2014 and earned his Ph.D in physics (Neutrinoless Double Beta Decay) from Babasaheb Bhimrao Ambedkar University (A Central University), Lucknow. He has more than three years of teaching experience at graduate and postgraduate level. His research interests include Nuclear and Particle Physics, Neutrino Physics, Physics beyond the standard model and Optics.

Gamma function, Probability Integral, Probability Theory, Probability Fresnel Integrals, Hypergeometric Equation, Legendre Functions, Legendre's Equation, Bessel Functions, Orthonormality, Generating Function, Hermite Functions, Rodrigues Formula, Hypergeometric Equation, Associated Legendre Functions, Associated Laguerre Polynomials