Page 9 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2. Recurrence relations are quite useful as they help in finding whole class of J (x) in terms of
n
two or three J (x) of lower values of n i.e., n = 0, 1, 2.
n
3. Generating function for J (x) is introduced so that certain formulas involving Bessel
n
functions can be deduced. With the help of generating functions we can deduce recurrence
relations or certain other formulas straight away.
4. Finally we also discuss the zeros of Bessel functions as they will lead us to the completeness
,
as well as orthogonality properties of Bessel s Functions.
,
1.1 Bessel s Differential Equations from Laplace Equations

In dealing with the theory of potential problems in electrostatics or in gravitational field we
commonly use Laplace equations

2 2 2
V V V
= 0 ...(1)
x 2 y 2 z 2
Here V is a function of the Cartesian Co-ordinates. Any solution V of this equation, which is a
n
homogeneous polynomial of degree n in x, y, z is called the solid spherical Harmonies.
Depending upon the symmetry of the problem we can express Laplace equation in cylindrical
co-ordinates (r, , z) or spherical polar co-ordinates (r, , ). You must be knowing that the
relations between x, y, z and r, , z are
x = r cos

y = r sin ...(2)
z = z
Also the relation between x, y, z and r, , are

x = r sin cos
y = r sin sin ...(3)
z = r cos
,
1.2 Bessel s Differential Equations

, ,
To define Bessel functions we first of all obtain Bessel s Differential equation from Laplace s
,
equation. To do that we write Laplace s equations (1) in cylindrical co-ordinates as
2 2 2
V 1 V 1 V V = 0 ...(4)
r 2 r y r 2 2 z 2
We assume that V as a function of r, and z can be written as
, ,
V = R(r) ( ) z (z) ...(5)
, ,
Where R, , Z are functions of r, , z alone respectively. Substituting in (4) we get
2
2
d R 1 dR RZ d 2 d Z
Z 2 Z 2 2 R 2 = 0
dr r dr r d dz
2
2
1 d R 1 dR 1 1 d 2 1 d Z
Or 2 2 2 2 = 0 ...(6)
R dr rR dr r d Z dZ

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