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.

 

[CellCoree]

A Laurent series is a powerful tool in complex analysis because it allows us to approximate a function in a larger region than a Taylor series. While a Taylor series can only approximate a function in a neighborhood around a specific point, a Laurent series can approximate a function in a larger annulus-shaped region, including points outside of the radius of convergence of the Taylor series.

This is particularly useful when dealing with functions that have singularities, such as poles or branch points, as these cannot be approximated by a Taylor series. In these cases, a Laurent series can provide a more accurate and comprehensive representation of the function.

Furthermore, the Laurent series also allows us to study the behavior of a function near its singularities, providing valuable insights into the structure and properties of the function.

In summary, the purpose of a Laurent series in complex analysis is to provide a more complete and accurate approximation of a function, particularly when dealing with singularities, and to allow for a deeper understanding of the behavior of the function near these singularities.