# Complex analysis fun!

1. Feb 19, 2008

### Mystic998

1. The problem statement, all variables and given/known data
Show that $$\frac{z}{(z-1)(z-2)(z+1)}$$ has an analytic antiderivative in $\{z \in \bold{C}:|z|>2\}$. Does the same function with z^2 replacing z (EDIT: I mean replacing the z in the numerator, not everywhere) have an analytic antiderivative in that region?

2. Relevant equations
Um lots of things I imagine.

3. The attempt at a solution
Well, I'm pretty sure that I can do a partial fraction decomposition in both cases, then the appropriate logarithms would give me a function that's analytic on the region minus whatever line I do the branch cut on. But unless there's some huge typo in the problem, I don't think that's what's being sought. I'm not really sure what else to do in this situation though. I have some other thoughts on the problem that may or may not work, but they're kind of long winded, and I'd rather not go into them unless I really have to. So, any suggestions?

Last edited: Feb 19, 2008
2. Feb 19, 2008

### Mystic998

Well, I still haven't been able to come up with anything. I thought maybe I could use specific branch cuts of log to show that the partial fraction decomposition integrates to zero around any closed curve in the region (I think it would require different branch cuts for each integral, so I don't know how valid this is), but then I have a problem if the closed curve has the complement of my region "inside" it.