Page 33 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes J ( )
x
Using (v), lim n n 0
x x
Using (iv) and (vi), (iii) reduces to

1
n
x
x J n 1 ( )dx
2 n (n 1)
0
1.8 Summary

 Bessel Differential equation is seen to have x = 0 as regular singular point
 x = is irregular singular point of the Bessel Differential.

 Bessel Differential equation is deduced from Laplace equation.
 Bessel Differential equation is of Fuchs Type and so Frobenius method of expanding
,
solution of Bessel s equation as power series in x is valid.
 The generating function of Bessel function is given by

x 1 n
t t J ( )
x
e z t = n
 With the help of generating function we obtain recurrence relations
x
 It is seen that J ( ) does not have repeated zeroes except at x = 0.
n
1.9 Keywords

Ordinary point of a Differential equation is such that the solution can be expressed in terms of
a power series.
Regular singular point x = x is such that p(x), q(x) of the differential equation
0
2
d y dy
x
p ( ) q ( )y 0
x
dx 2 dx
behave as
2
x x p x finite lim , x x 0 q x = finite lim
0
x 0 x x 0 x
Recurrence relation is a relation involving a few Bessel functions i.e. it involves
dJ ( )
x
J n ( ), J n 1 ( ), J n 1 ( ) and n .
x
x
x
dx
Generating function is such a function which on expansions gives the values of J ( ).
x
n
Fuchs type differential equation satisfies the properties as given above.
Indicial equation gives the values of the parameter appearing in power series expansion of
J n ( ).
x

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