On the bottom of the page 399. of Iwaniec and Kowalski's Analytic Number Theory, the authors claim that $$h(t)=\int_{\mathbb H}k(i,z)y^s d\mu (z)$$ yields $$h(t)=\sqrt{\pi}\frac{\Gamma(s\frac{1}{2})}{\Gamma(s+1)}R^s +O(R^{1/2}+S)$$ for $\frac{1}{2} < s\le 1$, where the implied constant depends on $s$. Can somebody show me at least the main steps of establishing the second equation from the first one?

Just a hint what is expected to be done here. The point pair invariant $k(w,z)$ comes from the function $k:\mathbb{R}^+\to\mathbb{C}$ such that $k(u)=1$ for $u\leq(R2)/4$, $k(u)=0$ for $u\geq(R+S2)/4$, and $k(u)$ is linear in between. The integral $h(t)$ is the Selberg transform of this function according to Lemma 15.6. Here we use that the function $y^s$ has Laplacian eigenvalue $s(1s)$ with $s=1/2+it$, and at $y=1$ it equals $1$. Now the task is to approximate $q(v)$, $g(r)$, $h(t)$, in this order, as given in Lemma 15.6. 

