Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$: $$ F(z)=\bigg(\alpha-i\frac{\sqrt{z}}{4\pi}\bigg)^{-1}\int_{\mathbb{R}^3}dx\bigg( f(x)\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}\bigg)$$ ? I have tried to evaluate $$F(z)-F(z')$$ using the theorem of dominated convergence.
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