= Solution
For planar <incompressible flow> and <irrotational flow>, combine a <velocity potential> and a <streamfunction> into the <complex potential of a two-dimensional potential flow>:
$$
\boxed{w(z)=\phi(x,y)+i\psi(x,y),\qquad z=x+iy.}
$$
The chosen conventions give
$$
u=\phi_x=\psi_y,\qquad v=\phi_y=-\psi_x.
$$
These are the <Cauchy-Riemann equations>, so $w$ is a <holomorphic function> wherever the flow is regular and a consistent local potential is chosen. Its <complex derivative> is the <complex velocity>
$$
\boxed{\frac{dw}{dz}=\phi_x+i\psi_x=u-iv.}
$$
A spatially <uniform flow> has $w=(U-iV)z+$ a constant. The following <contour integral> identities also apply to nonuniform <potential flow>, including the specified <line vortex>. A <complex potential of a two-dimensional potential flow> may be multivalued around a source or vortex even though its derivative is single-valued.
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