Question

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

Answer 1

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.