How to calculate the converge radius of a Laurent series

[kakarotyjn]

A method to get the Laurent series of a complex function is by undetermined coefficient.For example f(z)=cot(z)=cos(z)/sin(z).If we want to get the Laurent series of cot(z),we can expand cos(z) and sin(z) to Taylor series respect,then assume the series of cot(z) is a_{ - 1} z^{ - 1} + a_0 z^0 + a_1 z^1 + ...,

we can get a_-1,a_0... one by one.

But how to calculate its convergence radius?

Thank you!

 

[Gib Z]

Just as regular power series are valid in the disc of largest radius that doesn't contain a singularity, Laurent series are valid in the largest annulus that doesn't contain any singularities.