This is a question that has come up in the Physics of Gravitational Microlensing but I feel it is more suited for a mathematics forum. The equation relating light paths in the source plane caused by masses in the complex plane labelled $z_k$ is $$z_{s}=z-\sum_{k=1}^{N} m_{k} \frac{z-z_{k}}{\left(z-z_{k}\right)\left(\bar{z}-\bar{z}_{k}\right)}=z-\sum_{k=1}^{N} \frac{m_{k}}{\left(\bar{z}-\bar{z}_{k}\right)}$$.
Now, I have the equation for the magnification of light from the source plane as $\mu=\frac{1}{\operatorname{det} \mathrm{J}}, \quad \mathrm{J}=\frac{\partial\left(x_{s}, y_{s}\right)}{\partial(x, y)}$ and I want to derive the equality $$\operatorname{det} \mathrm{J}=1-\frac{\partial z_{\mathrm{s}}}{\partial \bar{z}} \frac{\overline{\partial z_{\mathrm{s}}}}{\partial \bar{z}}$$
The problem I am having is that I can compute $\frac{\partial z_{5}}{\partial \bar{z}} \frac{\overline{\partial z_{5}}}{\partial \bar{z}}$ to be: $$\left(\frac{\partial z_{s}}{\partial z}\right) \overline{\left({\frac{\partial z_{s}}{\partial z}}\right)}=\left(\sum_{k_{1}=1}^{N}\frac{m_{k_1}}{\left(\bar{z}-\bar{z}_{k_1}\right)^{2}}\right)\left(\sum_{k_{2}=1}^{N} \frac{m_{k_2}}{\left(z-z_{k_2}\right)^{2}}\right)$$
But then, using $z=x+iy$, $z_k = x_k+iy_k$ I find: $$\begin{array}{l}{\quad x_{s}+i y_{s}=x+i y-\sum_{k=1}^{N} \frac{m_{k}}{\left(x-i y-x_{k}+i y_{k}\right)}} \\ {\Rightarrow \frac{\partial x_{s}}{\partial x}=1-\sum_{k=1}^{N} \frac{m_{k}}{\left(x-i y-x_{k}+i y_{k}\right)^{2}}=1-\sum_{k=1}^{N} \frac{m k}{\left(\bar{z}-\bar{z}_{k}\right)^{2}}}\end{array}$$
And for every other derivative in the Jacobian I find the same expression $\left(\sum_{k=1}^{N} \frac{m_{k}}{\left(z-z_{k}\right)^{2}}\right)$ either multiplied by $1,-1$ or $i,-i$ plus a constant. Since the expression $\left(\frac{\partial z_{s}}{\partial z}\right) \overline{\left(\frac{\partial z_{s}}{\partial z}\right)}$ contains the conjugate of $\bar{z}$ or just $z$ these won't be equal. I am unsure what errors I am making here.