\[\operatorname{arctan}z=\tfrac{1}{2}\operatorname{arctan}\left(\frac{2x}{1-x^{2%
}-y^{2}}\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}%
\right),\]