Holomorphic 1/z

I haven't really looked at your work yet (since I've just taken two finals and have my physics final tomorrow), but I swear I will get around to looking at it soon. My questions for now are:

1. Does the branch cut always have to be of length 2π?
2. What if the function wraps around itself more than once within this 2π interval? I don't have an example, but I'm sure there exists complex functions that wrap around themselves more than once within [0,2π.
3. Not really related to anything, but I'm looking over the complex argument function, which is really just arctan(v/u) (or is it u/v?). Does it have a winding number of 2π, because it satisfies the Cauchy-Riemann equations, or is it just because of the singularity? Does satisfying the Cauchy-Riemann equations denote that a function has a singularity?
4. Can I imagine the value of the singularity as some sort of height between two branches? (like the distance between two leaves of a helicoid)
5. For what function would I get a value that's not 2πi, and all the constants are 1? (for example, not 2/z)

Gah, /dredging memories. I wish I had a complex book, my prof just went with his own notes.

1. No, a cut has infinite length - it extends from your singularity to any point at infinity and you choose how it gets there. Make use of that.
2. e^ixθ, x > 1, wraps around more than once. You still only choose 1 branch cut, but the interval shrinks. Say you choose re^i2θ in your parametrization... Then run θ from [0,π to keep it single valued. (not sure if help/answer)
3. arctan has an infinite line of singularities every umm... π along the imaginary axis. It wraps around every 2π because sin and cos do... Satisfying the C-R equations implies a function is analytic, nothing about singularities. There isn't really an equation to satisfy that'll imply singularities, you have to notice that you divide by 0 for some value in the plane.
4. Try the height between 2 wraparounds, I think that's applicable... (I never felt the need to attach a visual to the strength of a singularity :f6
5. There may be some pathological example that I don't know about. But usually, it's 2πi. Also, for the wording of this question... I think you want to mean parametrization instead of function? Any function with a singularity can have that point circled by a path, which naturally leads to complex exponentials in the parametrization. I think you would need a geometry other than the complex plane to produce such results, and even then...