Where is it analytic

[Bach-hand]

Can anyone describe the largest region in which g(z) = 1/[Log(z)-(i*pi)/2] is analytic?
The question is very different from anything we have done in our lessons and has me at a loss.

 

[HallsofIvy]

It couldn't be very different. I presume you know that log(z) is analytic for all z except the negative real axis. And that 1/z is is analytic for all z except 0. For what z are both of those true?

 

[Bach-hand]

Then the function is not analytic at Log(z) = i*pi/2 and when z = pi (due to the branch cut) ?