I want to compute the velocity field induced by a vortex sheet, connecting the two points 1 and $z_B$, in the complex plane $z=z_1+i z_2$, see Fig. 1 below. The complex conjugate of the velocity field is given by
$u^*= -\frac{i}{2 \pi} \int_1^{z_B} \frac{dz'}{z-z'}$.
Evaluation of the integral and parametrization of the upper integration limit to $z_B=1+\exp(i \beta)$ leads to
$ u^*= \frac{i}{2 \pi} \ln\left( \frac{z-1-\exp(i \beta)}{z-1} \right)$.
Sketch of the vortex sheet (red) in the complex plane
Everything works as expected for $\beta=0$, see Fig. 2.
However, if I set $\beta$ to different values ($\beta \in [0, \pi/2]$) I get some troubles, see Fig. 3 for $\beta=\pi/4$ and Fig. 4 for $\beta=\pi/2$. Note, that I used the default log() and conj() functions from Matlab for computing the natural logarithm and the complex conjugate respectively.
My conclusion is, that I am facing some troubles with the branch-cuts. My quick-and-dirty solution so far is to evaluate $u^* \exp(-i \beta)$ instead of $u^*$, which gives the expected vortex sheet velocities. Since I am ultimately interested in evaluating more complex functions, I need to understand how to approach the described issue analytically.
I am no mathematician and I have only very little knowledge of complex analysis, so please apologize my ignorance. Thanks for any help!
I finally solved my problem by realizing that my expressions works well for $\beta=0$. Now, instead of turning the vortex sheet by an angle $\beta$, I turned the points at which I am interested in the induced velocity by $-\beta$: $y=(z-1)\exp(-i \beta)+1$. Since a vortex sheet is essentially a branch cut, which defines the jump in the tangential velocity component, I had to turn the result by another $-\beta$. In doing so, the complex conjugate of this expression yields the correct velocity field:
$u^*= \frac{i}{2 \pi} \ln\left( \frac{y-2}{y-1} \right) \exp(-i \beta) $