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