How to evaluate these complex integrals? I am trying to learn string theory in AdS. Following this review:
 https://arxiv.org/abs/1104.2604, I have come across the following integrals when trying to calculate some OPEs:

$$
I = \int_{\mathbb{C}} \frac{ d^{2} u }{(u - w)^{2} (u - z)}.
$$ 

The author claims on page 137, equation (D.4), that 

$$
I = - 2 \pi \frac{\bar{z} - \bar{w}}{(z - w)^{2}}.
$$

How is this possible? I have no idea how to compute this, or even how to
search for it in the literature. Any help is appreciated.
I have found a way. It is likely to be incorrect, but it reproduces all integrals of the paper.

Define 

$$
I  = \int \frac{d^{2} u}{(u - w)^{2} (u -z)}
$$

and take

$$
\frac{\partial I}{ \partial \bar{z}} = - 2 \pi\int \frac{d^{2} u}{(u-w)^{2}} \delta^{2} (u-z) = - 2 \pi \frac{1}{(z - w)^{2}}.
$$

The same way, compute

$$
\frac{\partial I}{\partial \bar{w}} = -2 \pi \frac{\partial}{\partial w}\int \frac{d^{2}u}{(u-z)} \delta^{2} (u - w) =   2 \pi \frac{1}{(z - w)^{2}}.$$

It follows then

$$
I = - 2 \pi \frac{\bar{z} - \bar{w}}{(z-w)^{2}}.
$$

Notice that

$$
\frac{\partial}{\partial \bar{z}} \left[ \frac{\bar{z}}{(z-w)^{2}} \right]
=  \frac{1}{(z-w)^{2}} - \bar{z} \frac{\partial}{\partial z}\, \delta^{2} (z-w) = \frac{1}{(z-w)^{2}}
$$

because the delta multiplies the odd function $\bar{z}$.

If someone knows a way to obtain this result in any other way, please post it.
I have found a way. It is likely to be incorrect, but it reproduces all integrals of the paper.

Define 

$$
I  = \int \frac{d^{2} u}{(u - w)^{2} (u -z)}
$$

and take

$$
\frac{\partial I}{ \partial \bar{z}} = - 2 \pi\int \frac{d^{2} u}{(u-w)^{2}} \delta^{2} (u-z) = - 2 \pi \frac{1}{(z - w)^{2}}.
$$

The same way, compute

$$
\frac{\partial I}{\partial \bar{w}} = -2 \pi \frac{\partial}{\partial w}\int \frac{d^{2}u}{(u-z)} \delta^{2} (u - w) =   2 \pi \frac{1}{(z - w)^{2}}.$$

It follows then

$$
I = - 2 \pi \frac{\bar{z} - \bar{w}}{(z-w)^{2}}.
$$

Notice that

$$
\frac{\partial}{\partial \bar{z}} \left[ \frac{\bar{z}}{(z-w)^{2}} \right]
=  \frac{1}{(z-w)^{2}} - \bar{z} \frac{\partial}{\partial z}\, \delta^{2} (z-w) = \frac{1}{(z-w)^{2}}
$$

because the delta multiplies the odd function $\bar{z}$.

If someone knows a way to obtain this result in any other way, please post it.