This is a brief part of an exercise related to differential geometry on the minimal surface Scherk. However I am stuck in an analytical part.
Let $$F(x,y)= \arg\left(\frac{z+i}{z-i}\right)$$ where $z = x + iy$ with $z$ different from $\pm i$, and $\arg$ is the argument of the point with the real axis.
Let $$\phi(z) = \frac{\partial F}{\partial x} - i\frac{\partial F}{\partial y}$$
Show that $\phi$ is analytic.
Any hints on substitutions or steps to find out what are $\partial_{x}F$ and $\partial_{y}F$? ?
Let $D$ be any disk not containing the points $i$ or $-i$. The function $(z+i)/(z-i)$ is holomorphic and non-zero in $D$, and therefore has a complex logarithm in $D$. Let $U$ and $V$ be the real and imaginary part of that holomorphic logarithm: $$ f(z) = \log \frac{z+i}{z-i} =U(z) + i V(z) \, . $$ Differentiation and using the Cauchy-Riemann equations gives $$ f'(z) = \frac{1}{z+i} - \frac{1}{z-i} = U_x(z) + i V_x(z) = V_y(z) + i V_x(z) \, . $$ $F$ is (apart from a possible constant multiple of $2\pi$) the imaginary part of $f$, so that $$ \phi(z) = F_x(z) - iF_y(z)= V_x(z) -iV_y(z)\\ =-if'(z) = \frac{-2}{z^2+1}\, . $$