= Solution
For the <upper half-plane (complex analysis)>[upper half-plane] use $0<\theta<\pi$, and interpret the source's <arctangent> as this <argument>. Then $V=(V_0/\pi)\operatorname{Im}(z\operatorname{Log}z)$ is <harmonic>, so $\mathbf E=-\nabla V$ has zero curl and divergence where there is no <Electric charge>[charge]. On the positive boundary $\theta\to0$ and $y\log r\to0$; on the negative boundary $\theta\to\pi$. Hence the two <Boundary value>[boundary values] are respectively zero and $V_0x$.
\b[Branch qualification:] the principal real <arctangent> alone gives zero on \i[both] sides of $y=0$, contradicting the stated negative-axis <boundary condition>, for example at $x=-1$. The continuous upper-half-plane <argument> repairs this notation; it is not an optional <additive constant>.
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