LAURENT SERIES
Welcome to the blog guys, This post here is about a very important concept in the complex-plane calculus namely LAURENT SERIES, eponymous to a french mathematician named 'PIERRE ALPHONSE LAURENT' as he published it in 1824.
INTRODUCTION
To begin with, Laurent series is nothing but a generalization of the famous Taylor series. That said, one might ask- ain't Taylor series already a generalization of Maclaurin series?? WELL YES IT IS!! so what do we need a further generalization for??
As we know, Taylor series are centered at a complex constant denoted by Z0, which by definition, can be any point in the domain D of a function f(z) and the latter has to be analytic in the whole D including at Z0 of course. This way Taylor series easily represent entire & analytic functions in the non-negative powers of Z-Z0. And that is the limitation here, as you can not represent functions, with some singularities, in powers of Z-Z0 using a Taylor series and that's where lies the need to generalize and that's where Laurent series comes in.Laurent series incorporates negative powers of Z-Z0 along with the non-negative ones and unlike Taylor series, converges in an annulus centered at Z0,in the hole of which singularities may lie.
THEOREM
Let f(z) be analytic in a domain containing circles C1 and C2 and the annulus between them,then f(z) can be represented by Laurent series as,
where the coefficients are given by,
taken counter-clockwise around any simple closed path C that lies in the annulus and encircles the inner circle. This series converges and represents f(z) in an enlarged open annulus.
THE PRINCIPAL PART
In a special case where the only singularity is at Z = Z0, the series(or finite sum) of the negative powers of Z-Z0 is called the principal part of f(z).
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