= Solution
Choose a continuous <argument> $\theta$ on a <simply connected domain> avoiding zero, and put $r=\sqrt{x^2+y^2}$. With $u=x\log r-y\theta$, direct <differentiation> gives
$$
u_x=1+\log r,\quad u_y=-\theta,\quad u_{xx}=x/r^2,\quad u_{yy}=-x/r^2.
$$
Here $\theta_x=-y/r^2$, $\theta_y=x/r^2$; these identities specify the branch needed in the written <arctangent>. Integrating the <Cauchy-Riemann equations> gives
$$
\boxed{f(z)=z\operatorname{Log}z+iC,\qquad v=y\log r+x\theta+C,\quad C\in\mathbb R.}
$$
The literal principal $\arctan(y/x)$ works only separately on suitable half-planes. It cannot stand for a single continuous <argument> across the <imaginary axis>.
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