Quote Originally Posted by Lenggries View Post
I was considering trying to construct a function over complex numbers whose output always produced positive real numbers
Of course, there are many, many such functions, eg. f(x + iy) = x^2 + y^2 + 1. If you were at all curious, the problem I was actually working on is more general and at least involves a complex-valued function. I'm not sure if you're familiar with Lebesgue integration, but the statement is essentially, "Suppose f is a complex measurable function on a measure space (X, M, u) where u is a positive measure. Let phi(p) = integral over X of |f|^p du for 0 < p < infinity. Let E be the set of p where phi(p) is finite. Show that E can be an arbitrary connected subset of (0, infinity)." I specialized it to sequence spaces (so u = counting measure, X = naturals, integral becomes sum) so that people with experience only in infinite sums might have something to say.

My real solution is actually a bit different from what I wrote, though I don't particularly like either example. I constructed a countable disjoint union measure space and a sort of change of variables operation which allowed me to combine E's by intersecting countably many of them. Starting with the sum 1/n series then gives the desired conclusion quickly. I don't like either solution that well--both are a bit ugly and tedious to write up, though the disjoint union one is perhaps less so since at least one gets a general construction out of the deal.

but since you already have your solution, I'll save myself the headache
If you ever want to spend a few more minutes at it, please feel free. Somehow I have the feeling that I'm ignoring a relatively simple example. Just writing the terms of my example explicitly is extremely messy.