Consider the hyperbolic plane $H=R\times(0,\infty)$ with metric $ds^2=\frac{dx^2+dy^2}{y^2}$.
$l(x+iy)=Arccosh(1+\frac{x^2+(y-1)^2}{2y})$ is the hyperbolic distance from $\sqrt{-1}$.
The conical function is $$ f_{-\frac{1}{2}}(\cosh l)=\int_0^1 (\cosh l +\sinh l \sin 2\pi x)^{-\frac{1}{2}} dx $$
Then how to calculate $\partial_yf$ ?
In fact, I don't understand it well. Seemly ，it is hard to get $f(x+iy)= ?$ Thanks for any answer or hint.