Page 564 - Advanced_Engineering_Mathematics o'neil

544 CHAPTER 15 Special Functions and Eigenfunction Expansions

With k = i,
y(x) = c 1 J 0 (ix) + c 2 Y 0 (ix)
is the general solution of
1

y + y − y = 0,
x
for x > 0. This differential equation is a modified Bessel equation of order zero, and J 0 (ix) is a
modified Bessel function of the first kind of order zero. Usually this is denoted I 0 (x):
1 2 1 4 1 6
I 0 (x) = J 0 (ix) = 1 + x + x + x + ··· .
2 2 2
2 2
2 2 2 4 2 4 6
Normally Y 0 (ix) is not used, but instead the second solution is chosen to be
1 2
K 0 (x) =[ln(2) − γ ]I 0 (x) − I 0 (x)ln(x) + x + ···
4
for x > 0. Here γ is the Euler constant and K 0 is a modified Bessel function of the second kind of
order zero. Figure 15.9 shows partial graphs of I 0 and K 0 .
For x > 0, the general solution of
1
2
y + y − b y = 0
x
is
y(x) = c 1 I 0 (bx) + c 2 K 0 (bx).
It is routine to manipulate the series for I 0 (bx) to obtain
x

xI 0 (bx)dx = I (bx) + c, (15.9)
0
b
for any nonzero b.
Sometimes we need to know how I 0 (x) behaves for large x. We can obtain an expression for
I 0 (x), valid for large positive x, as follows. Begin with the fact that I 0 (x) is a solution of
1

y + y − y = 0.
x

6

5

4

3

2

1

0
0 0.5 1 1.5 2 2.5 3
x
FIGURE 15.9 I 0 (x) and K 0 (x).

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:20 THM/NEIL Page-544 27410_15_ch15_p505-562