Laurent Series

Laurent Series:

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 ≤ r₁ ≤ r₂  . If the Laurent seriesconverges on D(P, r₂) \ to a function f, then, for any r satisfying r₁ < r < r₂, 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 ≤ r₁ ≤ r₂  and f : D(P, r₂) \   → C is holomorphic, then there exist complex numbers a_j such that,

converges on D(P, r2) \ to f. If r₁ < s₁ < s₂ < r₂, then the series converges absolutely and uniformly on D(P, s₂) \ . this below) that f ≡ 0.

The series expansion is independent of s₁ and s₂. In fact, for each fixed j = 0,±1,±2, . . . , the value of

is independent of r provided that r₁ < r < r₂.

Figure of Cauchy Integrals on an anulus: