# Section 7.3a Laurent Series Representations

D[R](a) but is analytic in the punctured disk D[R]^`*`*`(a)` ={`z: 0`;`` < ``;abs(z-alpha) < R;}. For example, the function f(z) = exp(z)/(z^3) is not analytic when z = 0 but is analytic for 0 < abs(z);. Clearly, this function does not have a Maclaurin series representation. If we use the Maclaurin series for g(z) = exp(z) however, and formally divide each term in that series by z^3;, we obtain the representation
f(z) = 1/(z^3); exp(z) = 1/(z^3) + 1/(z^2) + 1/2!/z + 1/3! + z/4! + z^2/5! + z^3/6! + `...`
which is valid for all z such that 0 < abs(z). This example raises the question as to whether it might be possible to generalize the Taylor series method to functions analytic in an annulus
A(alpha,r,R); = {`z: r`;`` < ``;abs(z-alpha) < R;}.
Perhaps we can represent these functions with a series that employs negative powers of z in some way as we did with f(z). As you will see shortly, this is indeed the case. We begin by defining a series that allows for negative powers of z.