Page 42 - Calculus with Complex Numbers

The Laurent expansion is used for functions which have a singularity at c. W e
classify singularities according to the type of Laurent exparlsion obtained. W e call
that part of the Laurent exparlsion with negative powers of z - c the principal
part. We say flz) has a pole at z = c if the principal part has only linitely many
non-zero terms. lf the principal part has inlinitely many non-zero terms we say
flz) has an essential singularity.
The order of a pole is the largest n for which tz-n # 0. A pole of order 1 is
called a simple pole. A pole of order 2 is called a double pole. The residue of flz)
at z = c is the coeflicient tz-l in the Laurent expansion at z = c.
For example, f (z) = clx has an essential singularity at z = 0, since the Laurent
expansion at z = 0 is

The residue of clx at z = 0 is 1 .
On the other hancl, g (z) = ez/z4 has a pole of order 4 at z = 0, since the Laurent
expansion at z = 0 is
ez 1 .:2 z3
.
V
T
F = V 1 + z + + + ' ' '
1 1 1 1 1 1

'
?
+ - + ' ' ' .

'
= + + T?- V c
?
The residue of ez/z4 at c = 0 is 1/3 ! = 1/6.
3 .9 C aI cu Iation of Laurent expansions
W e proceed by way of example. Corlsider the function
1
f U) = 1 + z ,
z
which has singularities at z = Ljzi .
W e lind the Laurent expansion at z = i by putting t = z - i and expanding in
powers of t . W e obtain

which shows that flz) has a simple pole at z = i with residue 1/2ï .