278 CHAPTER 7 • TAYLOR AND LAURENT SERIES
Another example is g (z) = co•.1-^1 , which bas an isolated singularity at the
point 0 because the Laurent series for g isIf we define g (0) = -4, then g will be analytic for all z.
ii. If f has a p ole of order k at a, the Laurent series for f is
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f (z) = I: Cn (z - a)" (valid for all z EA (a, 0 , R) ),
n=- kwhere c _ k ":/= 0. For example ,sinz 1 1 z^2 z^4
f (z) = - = - -- + - - - + · · ·
z^3 z^2 3! 5! 7!has a pole of order 2 at O.
If f has a pole of order 1 at a , we say that f has a simple pole at a. For
example,1 z 1 z z^2
f (z) = - e z = -z + 1 + - + - + 2! 3! · · · ,has a simple pole at O.iii. If infinitely many negative powers of (z -a) occur in the Laurent series,
then f has an essent ial singularity at a. For example,I ( )
2. 1 1 -1 1 - 3 1 - 5
z = z sm-; = z - 31 z + 5!z - 1!z + · · ·has an essential singularity at the origin.Definition 7 .6: Zero of order kA function f analytic in DR (a) has a z ero of order k at the point a iff
f (n) (a)= 0, for n = 0, 1, ... , k - 1, but / (k) (a) -:j: 0.
A zero of order 1 is sometimes called a simple zero.