Complex Analysis Branches I have a single valued function is defined as a branch of the multivalued function
$$f=\ln[z(z+1)] \tag{*}$$ on the complex plane with the segment $[-1,0]$ of the real axis removed, and require $$f(1)=\ln[2] +2\pi i \tag{**}$$
Show that * and ** and the proposed branch cut do not describe a single valued function. 
I am trying to trace the imaginary part of $f$. $\text{Im}(f)= \arg(z) + \arg(z+1)$. And it cant see to satisfy the condition **. When $z=1$. I'm getting $\arg(z)=0$ and $\arg(z+1) =0$. 

I suppose you mean $f(1)=\ln(2) + 2\pi i$ which is a consistent choice
for $f$ at $1$. As is mentioned the function is not single valued. When you go once around the cut the arg increases with $4\pi$. One way to see this is to take the derivative $f'(z)=\frac{1}{z} + \frac{1}{z+1}$ and note that if you choose a contour $\gamma$ winding  counter-clockwise around the cut then
$\oint_\gamma f\, dz=4\pi i$. 
