Let I have the following function,


$f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$


Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y) \le m$, $m$ is a finite real number.


And $Q\left( z \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits\_z^\infty {{e^{ - \frac{{{u^2}}}{2}}}du} $.


I want to show that $f$ is analytic on the bounded complex plane.


\*\* Note: I know that the Q function is analytic on the real line. Does that information help while proving the analyticity of $f$? Any suggestions will be helpful.



Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?
The answer is: of course not. Indeed, take any real $a\ne0$, any real $b$, and 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 real and imaginary parts of the function
$$\mathbb R^2\ni(s,t)\mapsto g(s,t):=Q(as)Q(bt)\,\ln\big(Q(as)Q(bt)\big)\in\mathbb C$$
would satisfy the [Cauchy–Riemann equations](https://en.wikipedia.org/wiki/Holomorphic_function#Definition) at $(s,t)=(0,0)$ -- which the do not, because the partial derivative of $\Re g(s,t)$ in $s$ at $(s,t)=(0,0)$ is $a\,\dfrac{\ln(4/e)}{2\sqrt{2\pi}}\ne0$, while the partial derivative of $\Im g(s,t)[=0]$ in $t$ is $0$ everywhere.


