$H=R\times(0,\infty)$ is the hyperbolic plane. $\omega= i$ and $z=x+iy$. $d_H$ is the distance function of $H$.Consider $$ l=d_H(z,\omega)=\ln \dfrac{|z-\overline \omega|+|z-\omega|}{|z-\overline \omega|-|z-\omega|} =\ln\dfrac{\sqrt{x^2+(y+1)^2}+\sqrt{x^2+(y-1)^2}}{\sqrt{x^2+(y+1)^2}-\sqrt{x^2+(y-1)^2}} \\ f(\cosh l)=\int_0^1（\cosh l+\sinh l \sin 2\pi x）^{-1/2+ic} dx $$ $c$ is a constant. If the Laplace operator is $\Delta=y^2(\partial_{xx}+\partial_{yy})$, then , how to show $$ \Delta f(\cosh l)=-\bigg(\frac{1}{4}+c^2\bigg)f(\cosh l) $$
I try to use Maple to calculate it . But fail.
In fact, I am not sure whether the distance function is right .