# Zero velocity field inside an ellipse

I'm investigating the velocity field induced by a continuous distribution of 2D vortex points distributed along an ellipse $\{a\cos\theta,b\sin\theta\}$. I'm interested in the field inside the ellipse, and I need some help to prove whether this field is zero or not.

The intensity of each vortex point is proportional to $d\theta$ and not to the length along the ellipse. A vortex point located at a point $\boldsymbol{x'}$ induces a velocity field $\boldsymbol{u}(\boldsymbol{x})=\frac{d\theta}{2\pi |\boldsymbol{x}-\boldsymbol{x'}|} \boldsymbol{e}_\perp$ where ${e}_\perp$ is the unitary vector orthogonal to $(\boldsymbol{x}-\boldsymbol{x'})$ which is in 2D: $\boldsymbol{e}_z\times(\boldsymbol{x}-\boldsymbol{x'})/|\boldsymbol{x}-\boldsymbol{x'}|$. The total velocity field at a point $(x,y)$ inside the ellipse is obtained by integration over $\theta$. Numerical experiments seem to show that the field is zero inside the ellipse, but I cannot prove it. Dropping the factor $2\pi$, the field is in cartesian coordinates:

$$\boldsymbol{u}=\int_0^{2\pi} \left\{\frac{-y + b \sin\theta}{(x - a \cos\theta)^2 + (y - b \sin\theta)^2}, \frac{x - a \cos\theta}{(x - a \cos\theta)^2 + (y - b \sin\theta)^2}\right\} d\theta \stackrel{?}{=}\{0,0\}$$

Is the field really zero?

Maybe there is no need for the integrals to prove it. Maybe complex analysis is of help? I should mention that the following property is true in this problem: For any closed contour inside the ellipse the circulation is zero: $$\oint \boldsymbol{u}\cdot\boldsymbol{dl} =0$$

Since we are in a simply connected region, the velocity is the gradient of a potential which is single valued. But then is this potential constant?...

I'm able to prove that the velocity is zero on both the $x$ and the $y$ axis. Also $u_x$ is an even function of $x$ and an odd function of $y$. The opposite applies for $u_y$. I'm able to prove the result for a circle $a=b$. Can somebody help me for the ellipse? Any idea?