= Solution
{c}
For the <simple shear flow>, the vertical <vorticity> is
$$
\zeta=\partial_xv_y-\partial_yv_x=\boxed{-\beta}.
$$
For $\beta\ne0$, this is not an <irrotational flow>. A <complex potential of a two-dimensional potential flow> $w=\phi+i\psi$, analytic in the flow region, has <complex velocity> $w'=v_x-iv_y$. The <Cauchy-Riemann equations> imply both <incompressibility> and zero <vorticity>, so this shear has no such <complex potential of a two-dimensional potential flow>[complex potential]. It does have a <Cartesian streamfunction>, $\psi=\beta y^2/2$. In the degenerate case $\beta=0$, the resting fluid has a constant <complex potential of a two-dimensional potential flow>[complex potential].
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