Hello I am studying a paper named "Cycle integrals of the j functions and mock modular forms". I want to ask how we get equality which a marked as red in the image. Assumptions are
We are taking $Q=[a,b,c]$ be binary quadratic form with discriminant d>0 not a square and let $S_Q$ be the oriented semicircle defined by $a\tau^2 +bRe\tau +c=0$ for $\tau \in \mathbb{H}$ which is directed counterclockwise if a>0 and clockwise if a<0. For any $z\in S_Q$ let $C_Q$ be the directed arc on $S_Q$ from z to $g_Q z$ where $g_Q z= \frac{\frac{t+bu}{2}z+cu}{-au +\frac{t-bu}{2}}$ where t,u are smallest positive integral solutions of $t^2 -du^2=4$. Basically i want to ask how we changed integral to the semicircle $S_Q$ from the integration over arc $C_Q$. Any help would be appreciated.