= Complex potential of a two-dimensional potential flow
{title2=$W=\Phi+i\Psi,\quad W'=u-iv$}
For an incompressible irrotational two-dimensional <potential flow>, an analytic <complex potential of a two-dimensional potential flow> combines the velocity potential $\Phi$ and <streamfunction> $\Psi$. The <Cauchy-Riemann equations> give $u=\Phi_x=\Psi_y$ and $v=\Phi_y=-\Psi_x$, hence $W'(z)=u-iv$. Curves of constant $\Psi$ are <streamlines>. An analytic coordinate change $z=f(\zeta)$ transforms the complex velocity by $dW/dz=(dW/d\zeta)/f'(\zeta)$ wherever $f'\neq0$.
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