Annulus Of Convergence
Annulus of Convergence:
The set of convergence of a Laurent series is either an open set of the form {z : 0 ≤ r1 ≤ |z−P| ≤ r2}, together with perhaps some or all of the boundary points of the set, or a set of the form {z : 0 ≤ r1 ≤ |z −P| ≤ 
}, together with perhaps some or all of the boundary points of the set. Such an open set is called an (generalized) annulus centered at P. We shall let,
As a result, using this extended notation, all (open) annuli (plural of “annulus”) can be written in the form:
In precise terms, the “domain of convergence” of a Laurent series is given as follows:
be a doubly infinite series. There are unique nonnegative extended real numbers r1 and r2 (r1 or r2 may be 0 or ) such that the series converges absolutely for all z with r1 < |z − P| < r2 and diverges for z with |z − P| < r1 or |z − P| > r2 . Also, if r1 < s1 ≤ s2 < r2, then 
converges uniformly on {z : s1 ≤ |z−P| ≤ s2} and, consequently, converges absolutely and uniformly there. The reason that the domain of convergence takes this form is that we may rewrite the above series as:
From what we know about power series, the domain of convergence of the first of these two series will have the form |z − P| < r2 and the domain of convergence of the second series will have the form |(z − P)−1| < 1/r1. Putting these two conditions together gives r1 < |z − P| < r2.
Cauchy Integral Formula for an Annulus:
Suppose that 0 ≤ r1 < r2 ≤ and that f : D(P, r2) \ D(P, r1) ! C is holomorphic. Then, for each s1, s2 such that r1 < s1 < s2 < r2 and each z 
D(P, s2) \ 
it holds that,
