help for question on order of pole of function

****Let be a domain in , and let . It is given that a function

has a simple pole at . Consider the function given by

for all .

Is it true that has a double pole at ?

*This is my attempt.*

Since f has a pole of order 1 at . Then

where is analytic at and .

Since .

.

How do I proceed from here? Any idea or suggestion is welcome. Thanks in advance! :D

Re: help for question on order of pole of function

Re: help for question on order of pole of function

I'm sorry - I phrased that wrong.

The point is that