= Planar potential dipole
{title2=$w(z)=m/(z-a)$}
A <planar potential dipole> at $a$ with complex moment $m$ has <complex potential of a two-dimensional potential flow> $w(z)=m/(z-a)$. Its <complex velocity> is $w'(z)=-m/(z-a)^2$, interpreted as $v_x-iv_y$. Unlike the three-dimensional <potential dipole>, this is a two-dimensional singularity of the logarithmic-source potential. For example, two opposite source coefficients $s$ at $a\pm\delta/2$ have potential $s[\log(z-a-\delta/2)-\log(z-a+\delta/2)]$, whose leading term is $-s\delta/(z-a)$ as the separation tends to zero with $s\delta$ fixed. This gives the <planar potential dipole>[dipole] as a limiting source–sink pair, with the complex sign and orientation included in $m$.
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