Example 10.10.1.
Consider the conformal mapping
\begin{equation*}
w = f(z) = \frac{1}{\pi}(z^2-1)^\frac{1}{2} + \frac{1}{\pi}\mathrm{Log}\left[z+(z^2-1)^\frac{1}{2}\right]\text{,}
\end{equation*}
which we obtained by using the Schwarz-Christoffel formula. It maps the upper half-plane \(\mathrm{Im}(z) >0\) onto the domain in the \(w\) plane that lies above the boundary curve consisting of the rays \(u \le 0, \; v=1\) and \(u \ge 0, \; v=0\) and the segment \(u=0, \; -1 \le v \le 0\text{.}\) Furthermore, the image of horizontal streamlines in the \(z\) plane are curves in the \(w\) plane given by the parametric equation
\begin{align*}
w \amp = f(t+ic) =\frac{1}{\pi}(t^2-c^2-1+i2ct)^\frac{1}{2}\\
\amp + \frac{1}{\pi}\mathrm{Log}\left[ t+ic+(t^2-c^2-1+i2ct)^\frac{1}{2}\right]\text{,}
\end{align*}
for \(-\infty \lt t \lt \infty\text{.}\) The new flow is that of a step in the bed of a deep stream and is illustrated in Figure 10.10.2(a). The function \(w=f(z)\) is also defined for values of \(z\) in the lower half-plane, and the images of horizontal streamlines that lie above or below the \(x\) axis are mapped onto streamlines that flow past a long rectangular obstacle, which is illustrated in Figure 10.10.2(b).
