Principal branch of the logarithm

[mxmt]

Homework Statement

Define an analytic branch f(z) of w, such that f(z)=0 for the limit of z->\infty

Now what is f(1)?

Homework Equations

w=\frac{z+i}{z-i}

The Attempt at a Solution

The branch cut of the logarithm is: (-\infty,0)
All branches of the logarithm are:
f(z)=Log(z)+iArg(z)=Log(z)+2i\pik

But then f(1)=0, which is wrong.

 

[mxmt]

There was a typo in my first post:

w=\frac{z+i}{z-i}

Of course this should be w=log(\frac{z+i}{z-i})

Anybody who understands it now?

 

[mxmt]

A guess is that the line segment of z=(i,-i) is mapped onto the normal branch cut of the logarithm (-inf,0). Therefore, f(1)=exp(iPi/2) because this is where i is located in the complex plane.