7.3 • LAURENT SERIES REPRESENTATIONS 2677.3 Laurent Series Representations
Suppose that f (z) is not analytic in DR (a) but is analytic in the punctured
disk DR (a) = {z: 0 < lz - al< R}. For example, the function f (z) = ~e'
is not analytic when z = 0 but is analytic for lzl > 0. Clearly, this function
does not have a Maclaurin series representation. If we use the Maclaurin series
for g ( z) = e•, however, and formally divide each term in that series by z^3 , we
obtain the representation
1 z 1 1 1 1 z z^2 z^3f(z)=3e z =3+2+z z - (^21) .z +3,+4,.. +- (^51). +-6,. +···,
which is valid for all z such that lzl > 0.
This example raises the question as to whether it might be possible to gen-
eralize the Taylor series method to functions analytic in an annulus
A (a, r, R) = {z: r < lz -al < R}.
Perhaps we can represent these functions with a series that involves negative
powers of z in some way as we did with f (z) = ~e'. As you will see shortly, we
can indeed. We begin by defining a series that allows for negative powers of z.
Definition 7.3: Laurent seriesLet en be a complex number for n = 0, ±1, ±2, ± 3, .... The doubly infinite
00
series L; Cn (z - a)", called a Laure nt series, is defined by
n=- oo
00 00 00
L Cn (z - a)"= L C- n (z -a) - n +Co+ L Cn (z - a)", (7-21)
n.::- oo n=l n=l
provided the series on the right-hand side of this equation converge.
00
Remark 7.2 Recall that L: Cn (z -a)" is a simplified expression for the sum
n=O
00 00
co+ L; Cn (z -a)". At times it will be convenient to write L; Cn (z - a)"
n=l n=-oo
00 -1 00
as L: Cn (z - a)"= L; Cn (z - a)"+ L; Cn (z - a)" rather than using the
n=-oo n =-oo n=O
expression given in Equation (7-21). •