 Solution

On $D=\mathbb{C}\setminus [0,\infty )$ choose the continuous argument $0<\arg z<2\pi$, and putThis is holomorphic on the slit plane for every fixed $w$. The upper boundary value at $x>0$ is $x^w$, whereas the lower boundary value is $e^{2\pi iw}{x}^w$. This difference is exactly what the keyhole contour measures.