The answer is: of course not. Indeed, let $a=1$, $b=0$, and take any real $m>0$. If $f$ were analytic in $x,y$ such that $-m\le\Re x,\Re y,\Im x,\Im y\le m$ then it would be analytic in $x$ at $x=0$ for $y=0$. Then the function 
$$\mathbb R^2\ni(s,t)\mapsto\tfrac12\,Q(s)\ln(\tfrac12\,Q(s))$$
would satisfy the Cauchy–Riemann equations in a neighborhood of $(s,t)=(0,0)$ -- which it does not.