contour integral around $\infty$

Consider the function

where , the disc of radius centered at the origin, open, and . Show that is holomophic in an open subset of complex plane, containing the complement of the disc: and compute the integral

where is an integer.

I'm quite confused about this problem. Because, the $Log$ in the RHS should be the principal branch, i.e. , with . Then i should prove that does not intersect the real negative axis, but how to do this? However, suppose i could prove the holomorphicity, then i'm left to compute the integral, for which i tought to use the residue formula applied to point, so calling the integral,

where is the contour , so that . But then the computation of residue is very hard for me. Some help?