Page 71 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 61
Note that

∞

ν −t
(ν + 1) = t e dt (4.84)
0
and integrating by parts

∞

e dt = ν (ν)
−t ∞
(ν + 1) = [−t νe ] + ν t ν−1 −t (4.85)
0
0
and (4.82) can be written as
(n + ν + 1)
(1 + ν)(2 + ν)(3 + ν) ....(n + ν) = (4.86)
(ν + 1)

so that when ν is not an integer

∞ n
(−1)
2n+ν
J ν (x) = x (4.87)
2 2n+ν n! (n + ν + 1)
n=0
Fig. 4.3 is a graphical representation of the gamma function.
Here are the rules
1. If 2ν is not an integer, J ν and J −ν are linearly independent and the general solution of
Bessel’s equation of order ν is

u(x) = AJ ν (x) + BJ −ν (x) (4.88)

where A and B are constants to be determined by boundary conditions.

2. If 2ν is an odd positive integer J ν and J −ν are still linearly independent and the solution
form (4.88) is still valid.
3. If 2ν is an even integer, J ν (x)and J −ν (x) are not linearly independent and the solution
takes the form

u(x) = AJ ν (x) + BY ν (x) (4.89)

Bessel functions are tabulated functions, just as are exponentials and trigonometric functions.
Some examples of their shapes are shown in Figs. 4.1 and 4.2.
Note that both J ν (x)and Y ν (x) have an infinite number of zeros and we denote them as
λ j , j = 0, 1, 2, 3,...