Complex Variable Taylor Expansion at z=2i

[sachi]

I'm having trouble determining the order of the pole of

[exp(iz) - 1]/((z^2) + 4) at z=2i

I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ...
because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does Taylor's theorem still work for complex variables?)
thanks for your help.

 

[HallsofIvy]

It's pretty obvious, isn't it, that f(z)(z-2i)= (e^iz-1)/(z+2i) is analytic at z= 2i while f(z) itself is not. z= 2i is a pole of order 1.