Power series expansion for Log z

[smanalysis]

Homework Statement

Find the power series expansion of Log z about the point z = i-2. Show that the radius of convergence of the series is R = $$\sqrt{5}$$.

Homework Equations

None

The Attempt at a Solution

I know that Log z = (z-1) - (1/2)(z-1)^2 + (1/3)(z-1)^3 -...
So wouldn't this become (z-3-i) - (1/2)(z-3-i)^2 + (1/3)(z-3-i)^3 -... about the point z = i-2? What would a_k be for the series? Also, since values of Log z are unbounded in any neighborhood of the origin, I would think that this function cannot be extended analytically on any open disk containing the origin. But wouldn't it be analytic on the disk of radius |i-2| = |-2+i| = Sqrt((-2)^2 + 1^2) = Sqrt(5)? Then I would say that the radius of convergence would be Sqrt(5) for the disk centered at i-2, right? I would appreciate any suggestions, thanks in advance! :-)

 

[haruspex]

"about the point z = i-2" means set z=ω-(i-2) and find log(z) as a power series in ω.