Absolute convergence of the Liouville Upsilon defining integral

Fix a compact subset $K$ of the strip. Uniformly for $z\in K$, Taylor expansions at $t=0$ give $D_{\gamma }(t)=t^2\mathrm{/}4+O(t^4)$, ${\sinh }^2(at\mathrm{/}2)=a^2{t}^2\mathrm{/}4+O_K(t^4)$, and $a^2{e}^{-t}=a^2-a^{2}t+O_K(t^2)$. Hence ${\sinh }^2(at\mathrm{/}2)\mathrm{/}{D}_{\gamma }(t)=a^2+O_K(t^2)$ and the integrand of $J_{\gamma }$ is $-a^2+O_K(t)$; it is bounded uniformly near $0$.

From [F1], $\gamma \mathrm{/}4+1\mathrm{/}\gamma =Q\mathrm{/}2$. On $K$, compactness and the strict strip inequalities give $\eta :={\min }_{z\in K}(Q\mathrm{/}2-\mathrm{\mid }\mathrm{Re}a\mathrm{\mid })>0$. For $t\ge 1$, $D_{\gamma }(t)\ge c_{\gamma }{e}^{Qt\mathrm{/}2}$ for a constant $c_{\gamma }>0$, while $\mathrm{\mid }\sinh (at\mathrm{/}2){\mathrm{\mid }}^2\le e^{\mathrm{\mid }\mathrm{Re}a\mathrm{\mid }t}$. Thus the second term inside parentheses is bounded by $C_K{e}^{-\eta t}$; the first is bounded by $C_K{e}^{-t}$. After division by $t$, these bounds are integrable on $[1,\mathrm{\infty })$ uniformly for $z\in K$. Combining with step 1.1 proves the asserted absolute and locally uniform convergence.