# Laurent Series

**Laurent Series:
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A Laurent series on D(P, r) is a (formal) expression of the form

Note that the individual terms are each defined for all z D(P, r) \ { P }. The series sums from j = − to j = .

**Convergence of a Doubly Infinite Series:**

To discuss convergence of Laurent series, we must first make a general agreement as to the meaning of the convergence of a “P doubly infinite” series . We say that such a series converges if converge in the usual sense. In this case, we set;

In other words, the question of convergence for a bi-infinite series devolves to two separate questions about two sub-series.

**Uniqueness of the Laurent Expansion:**

Let 0 ≤ r_{1} ≤ r_{2 } . If the Laurent seriesconverges on D(P, r_{2}) \ to a function f, then, for any r satisfying r_{1} < r < r_{2}, and each j Z,

This claim follows from integrating the series term-by-term (most of the terms integrate to zero of course). In particular, the a_{j} ’s are uniquely determined by f.

**Laurent Expansions:**** **Now we may summarize with our main result:

**Theorem:** If 0 ≤ r_{1} ≤ r_{2} and f : D(P, r_{2}) \ → C is holomorphic, then there exist complex numbers a_{j} such that,

converges on D(P, r2) \ to f. If r_{1} < s_{1 }< s_{2} < r_{2}, then the series converges absolutely and uniformly on D(P, s_{2}) \ . this below) that f ≡ 0.

The series expansion is independent of s_{1} and s_{2}. In fact, for each fixed j = 0,±1,±2, . . . , the value of

is independent of r provided that r_{1 }< r < r_{2}.

**Figure of Cauchy Integrals on an anulus:
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