Largest Region Where g(z) = 1/[Log(z)-(i*pi)/2] is Analytic

[gestalt]

Homework Statement

Describe the largest region in which g(z) = 1/[Log(z)-(i*pi)/2] is analytic?

The Attempt at a Solution

Analytic is where the cauchy-euler equations hold, so I tried to take the partials to and set them equal so I could define a domain. I am not sure if the partials are correct.

I also tried to look at it from the following point of view. It would be analytic where the domain of g(z) is valid. My line of thinking is Log(z) is the principal log and it has a branch cut at pi. Thus pi is not in the domain. Also 1/z where z≠0 so Log(z)-(i*pi)/2 ≠ 0.
Log(z) ≠ (i*pi)/2. I think Log(i) = (i*pi)/2. If so then the parts not in the domain are pi and i.

Does either attempt sound like a possible solution?

 

[sunjin09]

branch point of Log[z] is 0, the branch cut can be arbitrarily chosen to go from 0 to infinity, so if you choose arg(z) in (pi/2,5pi/2) then g(x) is analytic everywhere except positive imag axis