Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2009 / B / 4 / iv / Solution

Separate the real and imaginary parts of the derivative and integrate from $D=(0,1)$:Because $\cos s\cot s=\csc s-\sin s$ and $\sin s\cot s=\cos s$, these integrals areAs $\theta \to 0^{-}$, $x\to +\infty$ and $y\to 1-2a/\pi$. This must equal the outgoing stream depth $a$, so

The solution has a vertical tangent at $D$ and approaches the horizontal line $y=a$ downstream, agreeing with both solid-wall and free-streamline conditions.