Why use a laurent series in complex analysis?

[ENgez]

In complex analysis, what exactly is the purpuse of the luarent series, i mean, i know that it apporximates the function like a taylor series, an if the function is analytic in the whole domain it simplifies into a taylor series. But i fail to see its purpose - what does it do that the taylor series can't?

 

[spamiam]

Hello!

In complex analysis, we're often interested in functions that are not holomorphic/analytic, but are meromorphic: http://en.wikipedia.org/wiki/Meromorphic_function.

In this case, Laurent series are a generalization of Taylor series, since now we can approximate functions that have poles using a series. Take a look at this article: http://en.wikipedia.org/wiki/Residue_(complex_analysis).

Also, just like power series, the collection of Laurent series has a rich algebraic structure as well.

 

[mathwonk]

why do we want to divide? the set of all quotients of taylor series are exactly the laurent series. i.e. laurent series are to taylor series as rational numbers are to integers.

 

[Landau]

But i fail to see its purpose - what does it do that the taylor series can't?

well, it has negative powers. E.g. you might want to consider rational functions, or e^/z/z, or e^(1/z), or ... just any quotient of holomorphic functions, as mathwonk says.