What is a Laurent series? Does it have a radius of convergence?

1 Answer
Steve M
Nov 18, 2016

For a Real based function f(x) where x in RR we can expand f(x) as a power series involving positive powers of x know as a Taylor Series.

f(x) = sum_(n=0)^oo a_n(x-a)^n where a_n= f^((n))(a)/(n!)

For a Complex based function g(z) where z in CC we can form a power series that also contains negative powers of z known as a Laurent series.

g(x) = sum_(n=-oo)^oo a_n(x-c)^n where a_n= 1/(2pii)oint_gamma f(z)/(z-c)^(n+1)dz

Here the a_n is a line integral in the Complex Plane.

A consequence of this is that a Laurent series may be used in cases where a Taylor Series is not possible.Both series have a radius of convergence.

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