Page 750 - Advanced_Engineering_Mathematics o'neil

730 CHAPTER 22 Singularities and the Residue Theorem

2. z 0 is a pole of order m, with m a positive integer, if c −m = 0, but
c −m−1 = c −m−2 = c −m−3 = ··· = 0.
3. z 0 is an essential singularity if c −n = 0 for infinitely many positive integers n.

Thus, z 0 is a removable singularity if the Laurent expansion about z 0 is actually a power
m
series, a pole of order m if 1/(z − z 0 ) is the largest power of 1/(z − z 0 ) appearing in the Laurent
expansion of f (z) about z 0 , and an essential singularity if the expansion of f (z) about z 0 has
infinitely many powers of 1/(z − z 0 ) with nonzero coefficients.

EXAMPLE 22.1

Let f (z) = (1 − cos(z))/z. Then f is differentiable for all z = 0 and is not defined at 0. Using
the Maclaurin series for cos(z), the Laurent expansion of f (z) around zero is

∞
1 − cos(z) 1 (−1) n
f (z) = = 1 − z 2n
z z (2n)!
n=0
1 1 2 1 4 1 6
= z − z + z − ···
z 2! 4! 6!
1 1 3 1 5
= z − z + z − ··· .
2! 4! 6!
Since this is a power series about 0, f has a removable singularity at 0. We can define f (0) = 0,
the value of this power series at z = 0, and this “extended” f is differentiable at 0.

EXAMPLE 22.2
2
Let f (z) = 1/(z + i) . This function is its own Laurent expansion about −i, where it is not
2
differentiable. Because the largest power of 1/(z + i) appearing in this expansion is 1/(z + i) ,
f has a pole of order 2 at −i. There is no way to define f (−i), so the extended function is
differentiable at −i.

EXAMPLE 22.3
5
Let g(z)=sin(z)/z . This function is differentiable except at z =0, where it is not defined. Using
the Maclaurin expansion of sin(z), the Laurent expansion of f about 0 is

∞ n 2n+1 ∞ n
(−1) z (−1)

f (z) = = z 2n−4
(2n + 1)! z 5 (2n + 1)!
n=0 n=0
1 1 1
= − + ···
z 4 3! z 2
for z = 0. The highest power of 1/z appearing in this expansion is 4, so f has a pole of order 4
at 0.

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-730 27410_22_ch22_p729-750