The Joukowski map $-\frac12(z+1/z)$ from the upper half-disk to the upper half-plane

Prove that \(f ( z ) = - \frac { 1 } { 2 } \left( z + \frac { 1 } { z } \right)\) is a conformal map from half disc \(\left\{ z = x + i y : \ | z | < 1 , \ y > 0 \right\}\) to upper half plane \(\mathbb { H } \overset { \cdot } { = } \left\{ z = \overset { \cdot } { x } + i y : \ y > 0 \right\}\)