Residue of exp(1/(z+i)): How to Find and Solve for z=-i

[aaaa202]

Hi, I want to know how you find the residue of z=-i for the function exp(1/(z+i)). Clearly, the function has an essential singularity at z=-i so the good ol' formula for the residue for a pole of order m, doesn't really work here. What do I do? :)

 

[clamtrox]

You expand the function as a Laurent series around the pole and recall that the residue at that point is the coefficient in front of term proportional to \frac{1}{z+i}

 

[aaaa202]

okay, can you help me how that is done. My book's section os Laurent series is quite poor. The only thing I know about expanding functions as them, is that you can sometimes use geometric series. You don't need to say how I should do it, just hint me at where to start.

 

[clamtrox]

Just expand e^x as a usual Taylor series, then plug in x=1/(z+i) and it's done!