Page 768 - Advanced_Engineering_Mathematics o'neil

748 CHAPTER 22 Singularities and the Residue Theorem

EXAMPLE 22.22
Let
1
F(z) = .
(z − 4)(z − 1) 2
2
Then F(z) has simple poles at ±2 and a double pole at 1. Compute
e tz 1
2t
tz
Res(e F(z),2) = lim = e ,
z→2 (z + 2)(z − 1) 2 4
e tz 1
tz
Res(e F(z),−2) = lim =− e −2t ,
z→−2 (z − 2)(z − 1) 2 36
and
d 2 −1 tz
tz
Res(e F(z),1) = lim ((z − 4) e )
z→1 dz
−2 tz
2
2
tz
−1
= lim(−2z(z − 4) e + te (z − 4) )
z→1
1 2
t
t
=− te − e .
3 9
Then
1 1 1 2
−1 2t −2t − te − e .
t
t
L [F](t) = e − e
4 36 3 9
22.4.1 Diffusion in a Cylinder
We will find the temperature distribution function for a homogeneous, solid cylinder of radius R
centered along the z-axis. This problem was solved in Section 17.5 using separation of variables.
We will now use the Laplace transform and Theorem 22.7 to obtain the temperature distribution
function. In the course of this, we will use properties of the modified Bessel function I 0 (x) and
the Bessel functions J 0 (x) and J 1 (x) of the first kind of orders zero and one, respectively. These
functions are developed in Section 15.3.
We will assume angular independence and use the heat equation in cylindrical coordinates.
The boundary value problem is
∂u ∂ u 1 ∂u
2
= + for 0 ≤r ≤ R,t > 0
∂t ∂r 2 r ∂r
u(r,0) = 0,u(R,t) = T 0 .
Apply the Laplace transform with respect to t to this problem to obtain
2
∂ U 1 ∂U
+ − sU(r,s) = 0.
∂r 2 r ∂r
This is a modified Bessel equation of order zero. A solution that is bounded at r = 0, which is
the center of the cylinder, is given by
√
U(r,s) = cI 0 ( sr),

where I 0 (z) = J 0 (iz). Transform the condition u(R,t) = T 0 to obtain U(R,s) = T 0 /s. Then
√ T 0
U(R,s) = cI 0 ( sR) = .
s

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 15:37 THM/NEIL Page-748 27410_22_ch22_p729-750