12. The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by
$$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
- Show that \(T\) maps points on the half-line \(\arg ( z ) = \frac { \pi } { 4 }\) in the \(z\)-plane into points on the circle \(| w | = 1\) in the \(w\)-plane.
- Find the image under \(T\) in the \(w\)-plane of the circle \(| Z | = 1\) in the \(z\)-plane.
- Sketch on separate diagrams the circle \(| \mathbf { Z } | = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane.
- Mark on your sketches the point \(P\), where \(z = \mathrm { i }\), and its image \(Q\) under \(T\) in the \(w\)-plane.