**Subject :**Math -3

**Unit :**Function of Complex variable

## Annulus of Convergence

**Annulus of Convergence:**

The set of convergence of a Laurent series is either an open set of the form {z : 0 ≤ r_{1} ≤ |z−P| ≤ r_{2}}, together with perhaps some or all of the boundary points of the set, or a set of the form {z : 0 ≤ r_{1} ≤ |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 r_{1} and r_{2} (r_{1 }or r_{2} may be 0 or ) such that the series converges absolutely for all z with r_{1} < |z − P| < r_{2 }and diverges for z with |z − P| < r_{1} or |z − P| > r_{2} . Also, if r_{1} < s_{1} ≤ s_{2} < r_{2}, 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| < r_{2} and the domain of convergence of the second series will have the form |(z − P)^{−1}| < 1/r_{1}. Putting these two conditions together gives r_{1} < |z − P| < r_{2}.

**Cauchy Integral Formula for an Annulus:**

Suppose that 0 ≤ r_{1} < r_{2} ≤ and that f : D(P, r2) \ D(P, r1) ! C is holomorphic. Then, for each s_{1}, s_{2} such that r_{1} < s_{1} < s_{2} < r_{2} and each z D(P, s_{2}) \ it holds that,