Velocity field induced by vortex points along 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?
 A: Yes, the velocity field inside the ellipse is really zero. To convince myself of it I have run a few numeric evaluations of the integral for various $(a,b)$ and $(x,y)$.
Here is how can we obtain this result analytically. 
First, we introduce the elliptical coordinate system with coordinates $\chi$, $\theta$: 
$$x = c \cosh \chi \cos \theta, \qquad y = c \sinh \chi \sin \theta, $$ so that the ellipse in question corresponds to a fixed value of $\chi$:
$$a = c \cosh \chi_0, \qquad b = c \sinh \chi_0,$$
we also assume $a>b$. 
Second, we compute the vorticity field. We start with single vortex
$$ \mathop{\mathrm{curl}} \left( \frac{\Gamma}{2\pi |\mathbf{x}-\mathbf{x}_0|} \mathbf{e}_\perp \right) = \mathbf{e}_z \Gamma \delta(\mathbf{x}-\mathbf{x}_0)=\mathbf{e}_z \Gamma \delta(x-x_0)\cdot\delta(y-y_0),$$ and integrate the expression over the ellipse:
\begin{multline}
\mathop{\mathrm{curl}} \mathbf{u} = \mathbf{e}_z \int_0^{2\pi} \delta(\mathbf{x}-\mathbf{x}_0(\theta')) d \theta' =  \mathbf{e}_z \int_0^{2\pi} \delta(x - a \cos \theta') \delta(y-b \sin \theta') d\theta' = \\= \mathbf{e}_z \int_0^{2\pi} c^{-2} (\cosh^2\chi-\cos ^2 \theta)^{-1} \delta(\chi-\chi_0)\delta(\theta - \theta') d\theta' = \\ = \mathbf{e}_z c^{-2} (\cosh^2\chi-\cos ^2 \theta)^{-1} \delta (\chi - \chi_0),
\end{multline}
where we used the expression for area element in elliptical coordinates: $dA = c^2 (\cosh^2\chi-\cos ^2 \theta)d\chi\, d\theta$.
Finally, we calculate the stream function $\psi$ with vorticity as a source: $$\Delta \psi = - \omega \equiv - \mathbf{e}_z \cdot \mathop{\mathrm{curl}} \mathbf{u}.$$ 
In elliptical coordinates this reads as:
$$
c^{-2} (\cosh^2\chi-\cos ^2 \theta)^{-1} \left( \frac{\partial^2 \psi}{\partial \chi^2}+\frac{\partial^2 \psi}{\partial \theta^2}\right) = - c^{-2} (\cosh^2\chi-\cos ^2 \theta)^{-1} \delta (\chi - \chi_0).
$$
Notice, that the factors on both sides are equal. So we could look for the solution depending only on $\chi$, compatible with $\mathbf{u}(\mathbf{0})=\mathbf{0}$: 
$$ \psi =  - H(\chi - \chi_0) \cdot (\chi - \chi_0), $$ wher $H(x)$ is Heaviside function. This shows, that inside the ellipse velocity field is indeed zero. It also enables us to calculate the velocity field outside the ellipse, right away we see that the streamlines will be confocal ellipses.
