I came across the following question and I'm having a hard time figuring out how to construct such conformal mapping.
Question: Find a conformal mapping $f$ which maps the cut plane $D_{1,\pi}$ onto the unit disc $D_1(0) = \{ z ∈ \mathbb{C} : |z| < 1 \}$. Recall that the cut plane $D_{1,\pi}$ is defined as $D_{1,\pi} = \mathbb{C} \setminus \{ z ∈ \mathbb{C} : z = 1 + re^{i\pi} = 1 − r, r \geq 0 \}$.
I know that the cut plane is describing a line lying on the real axis, starting at 1 and moving to the left where all values are negative, I'm however unsure how I might map this line to the disc of radius 1 and centred at 0.