Refer to Fig. 26, Appendix 2. As the point z moves to the right along the negative real axis, its image point w is to move to the right along the entire u axis. As z describes the segment 0 ≤ x ≤ 1 of...

Refer to Fig. 26, Appendix 2. As the point z moves to the right along the negative real axis, its image point w is to move to the right along the entire u axis. As z describes the segment 0 ≤ x ≤ 1 of the real axis, its image point w is to move to the left along the half line v = π i (u ≥ 1); and, as z moves to the right along that part of the positive real axis where x ≥ 1, its image point w is to move to the right along the same half line v = π i (u ≥ 1). Note the changes in direction of the motion of w at the images of the points z = 0 and z = 1. These changes suggest that the derivative of a mapping function should be

f  (z) = A(z − 0)
⁻¹(z − 1),

where A is some constant ; thus obtain formally the mapping function

w = π i + z − Log z,

which can be verified as one that maps the half plane Re z > 0 as indicated in the figure.