Can't prove this limit of complex numbers from a paper Okay so I found in a paper, marked as "simple exercise", the following thing: for $z,b \in \mathbb{C}$,
$$\lim_{b\to0} \frac{1}{z-b} + \frac{1}{2b} - \frac{1}{\overline{b}{\vphantom{b}}^2(z-\frac{1}{\overline{b}})} - \frac{1}{2\overline{b}} = z + \frac{1}{z} + Constant$$  
And no one has so far managed to confirm it. Can someone please help me.
 A: Equation (5) of the paper looks strange to me. It introduces complex potential of the ideal flow in the disk $|\zeta|<1$ with a point vortex at $\beta$, as $\frac{-i}{2\pi}W_0$ where 
$$W_0(\zeta,\beta)= \log\left(\frac{\zeta-\beta}{|\beta|(\zeta-\bar\beta^{-1})} \right) \tag5$$ 
This is a valid complex potential for the flow when $\beta\ne 0$. But the potential is not continuous at $\beta=0$ since the term $|\beta|/\bar \beta$ does not have a limit as $\beta\to 0$. It's no wonder we get something weird by differentiating it with respect to $\beta $ and letting $\beta\to 0$.
I think (5) should be replaced with 
$$W_0(\zeta,\beta)= \log\left(\frac{\zeta-\beta}{1-\bar\beta\zeta} \right) \tag{5*}$$ 
Equation (5*) is what we get by composing $\log \zeta$, the potential of vortex at the center, with the Möbius transformations that moves $\beta$ to the center. As $\beta\to 0$,   (5*) converges to $\log \zeta$ as we would expect. Since the difference between (5) and (5*) is a purely imaginary constant $\log(-\bar \beta/|\beta|)$, both potentials define the same flow. 
Using (5*) instead of (5),  the derivatives  come out nicer than in (10): 
$$\frac{\partial W_0}{\partial\beta} = -\frac{1}{\zeta-\beta},\qquad 
\frac{\partial W_0}{\partial\bar\beta} = \frac{\zeta}{1-\bar\beta \zeta} \tag{10*}$$
Hence, (11) becomes 
$$W(\zeta,\beta)= Ua\left(\frac{\partial W_0}{\partial\bar\beta}-\frac{\partial W_0}{\partial\beta}\right) = Ua\left(\frac{\zeta}{1-\bar\beta \zeta} + \frac{1}{\zeta-\beta}\right)  \tag{11*}$$
And now the limit $\beta\to 0$ gives the result that you expect. 
