I know that the radius of convergence of any power series can be found by simply using the root test, ratio test etc.
I am confused as to how to find the radius of convergence for an analytic $f$ such as
$f(z)=\frac{4}{(z-1)(z+3)}$.
I can't imagine that I would have to find the power series representation of this, find the closed form, and then use one of the convergence tests. I am fairly certain that the radius of convergence would have to do with the singularities at $1$ and $-3$, however, I can't find a formula for the radius of convergence..
Best Answer
It is very useful to remember that the radius of convergence of power series in the complex plane is basically the distance to nearest singularity of the function. Thus if a function has poles at $i$ and $-i$ and you do a power series expansion about the point $3+i$, then the radius of convergence will be $3$ since that is the distance from $3$ to $i$.