Complex potential

Remarks

The equations $\varphi_x = \psi_y$ and $\varphi_y = -\psi_x$ are the Cauchy–Riemann equations, which is why $w$ is holomorphic. Then $w' = \varphi_x + i \psi_x = u - i v$. Conversely, every holomorphic $w$ on $\Omega$ is the complex potential of a plane incompressible irrotational flow, with velocity $\overline{w'(z)}$. The potentials are determined up to additive constants.

Example: $w(z) = z^2/2$ has $\varphi = (x^2 - y^2)/2$ and $\psi = x y$. Then $w'(z) = x + i y$, so $(u, v) = (x, -y)$, the stagnation point flow, and indeed $u_x + v_y = 0$ and $v_x - u_y = 0$.

Non-example: the rigid rotation $(u, v) = (-y, x)$ has $v_x - u_y = 2 \ne 0$, so it has no complex potential. Indeed $u - i v = -y - i x = -i \bar z$, which is not a holomorphic function of $z$.

On a domain that is not simply connected, such as $\C \setminus \set{0}$, the complex potential can be multivalued; the point vortex has $w = \frac{\Gamma}{2 \pi i} \log z$.