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Richa Nandra, Lovely Professional University Unit 1: Bessel s Functions

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Unit 1: Bessel s Functions Notes

CONTENTS
Objectives
Introduction
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1.1 Bessel s Differential Equations from Laplace Equations
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1.2 Bessel s Differential Equations
1.3 On Second Order Differential Equation of the Fuchs Type
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1.3.1 Series Solution of Bessel s Differential Equation
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1.3.2 Solution of Bessel s Differential Equation when n is a Non-negative Integer
1.4 Recurrence Formulas for J (x)
n
1.5 Generating Function for J (x)
n
1.6 On the Zeros of Bessel Functions J (x)
n
1.7 Illustrative Examples
1.8 Summary
1.9 Keywords

1.10 Review Questions
1.11 Further Readings

Objectives

After studying this unit, you should be able to:
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 Deduce Bessel s Differential equation from Laplace equation
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 Obtain singular and non-singular points of Bessel s equations
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 Obtain series solutions of Bessel s equation by Frobenius Method
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 Establish recurrence relations between various Bessel s Co-efficient

 Obtain the formula for J (x) from its generating functions
n
 Obtain zeroes of Bessel Functions.
Introduction

In this unit we shall be dealing with the various forms of Laplace differential equation involving
Cartesian, Cylindrical and Spherical polar Co-ordinates.
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Bessel s functions play a very important and central place in optical phenomical and in applied
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mathematical process. Just as a Fourier series, power series, Bessel s functions are quite useful in
solving problems involving laplace equations in cylindrical co-ordinates. In this unit the
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importance is given to the following aspects of the Bessel s functions:
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1. Solution of Bessel s functions J (x), Y (x) for various values of n as well as for different
n n
expansions involving x or (1/x).

LOVELY PROFESSIONAL UNIVERSITY 1