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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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If you want braces to display, you need two backslashes like \\{ or use \lbrace and \rbrace (I changed this). I left the 'lonely' bracket in the display as it was not clear what you intended. –  quid Mar 6 '13 at 20:26
    
For the integral to be well-defined, what is the behavior of $f(x)$ near $x=0$? Or, is the differential operator $d$ applied to the whole thing? –  i707107 Mar 6 '13 at 21:30
    
I've modified the question!Is it clear now? –  Mario Mar 6 '13 at 22:18
    
The integral is well defined thanks to the Schwartz inequality –  Mario Mar 6 '13 at 22:26

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