Looking for an analytic mapping theorem

[Poopsilon]

Say we have a complex function f, analytic on some punctured open disk D\{a} where it has a pole at a. Is there some theorem which says something like: f must map D\{a} to a horizontal strip in ℂ of at least width 2π, or something like that?

 

[morphism]

There can be no such theorem. Think about what f(z)=1/z does in a punctured disc D\{0}: the image f(D\{0}) won't contain any strips.

What is true is that if f has a pole at a then you can find a sufficiently small open disc centered at a such that f(D\{a}) contains the complement of an arbitrarily large open disc. This essentially follows from studying the example f(z)=1/z above together with an application of the open mapping theorem.

 

[Poopsilon]

ok, thanks morphism.

 

[morphism]

I think I might've misinterpreted your question. In my first reading I assumed you were looking for strips centered about the x-axis, but now I see that you didn't make any such requirement. Anyway, as you can see from my previous post, f(D\{a}) will always contain arbitrarily wide strips, if you allow them to be 'high up' the y-axis.

 

[Poopsilon]

Ok that would work for me, although I cannot see how this follows from the open mapping theorem, could you explain further how such a result would be derived? Thanks.

 

[morphism]

Sorry for the late reply, been busy..

Anyway: consider g(z)=1/f(z) in D. Since f is holomorphic on D\{a} and has a pole at a, then g is holomorphic on all of D (i.e. the singularity at a is removable). Now apply the open mapping theorem to g.