Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / B / 6 / iv

Integrate real and imaginary parts from the lip value $z(-\pi )=i$:The identities $\sin s\cot (s/2)=1+\cos s$ andgive the explicit parametrizationAs $\theta \uparrow 0$, $x\to +\infty$ and $y\to 1-a$. This must be the downstream height $a$, soThe surface initially moves slightly left of the lip before turning right. Its leftmost point occurs at $\theta =-\pi /2$, with $x_{\min }=-(1-\log 2)/(2\pi )$. This is the half-contraction free streamline at a horizontal canal lip.