Question 2.15: Consider the transformation w = ln z. Show that (a) circles ...
Consider the transformation w = ln z. Show that (a) circles with center at the origin in the z plane are mapped into lines parallel to the v axis in the w plane, (b) lines or rays emanating from the origin in the z plane are mapped into lines parallel to the u axis in the w plane, (c) the z plane is mapped into a strip of width 2π in the w plane. Illustrate the results graphically.
We have w=u+i v=\ln z=\ln r+i \theta so that u=\ln r, v=\theta.
Choose the principal branch as w=\ln r+i \theta where 0 \leq \theta<2 \pi.
(a) Circles with center at the origin and radius \alpha have the equation |z|=r=\alpha. These are mapped into lines in the w plane whose equations are u=\ln a. In Figs. 2-17 and 2-18, the circles and lines corresponding to \alpha=1 / 2,1,3 / 2,2 are indicated.
(b) Lines or rays emanating from the origin in the z plane (dashed in Fig. 2-17) have the equation \theta=\alpha. These are mapped into lines in the w plane (dashed in Fig. 2-18) whose equations are v=\alpha. We have shown the corresponding lines for \alpha=0, \pi / 6, \pi / 3, and \pi / 2.
(c) Corresponding to any given point P in the z plane defined by z \neq 0 and having polar coordinates (r, \theta) where 0 \leq \theta<2 \pi, r>0 [as in Fig. 2-19], there is a point P^{\prime} in the strip of width 2 \pi shown shaded in Fig. 2-20. Thus, the z plane is mapped into this strip. The point z=0 is mapped into a point of this strip sometimes called the point at infinity.
If \theta is such that 2 \pi \leq \theta<4 \pi, the z plane is mapped into the strip 2 \pi \leq v<4 \pi of Fig. 2-20. Similarly, we obtain the other strips shown in Fig. 2-20.
It follows that given any point z \neq 0 in the z plane, there are infinitely many image points in the w plane corresponding to it.
It should be noted that if we had taken 0 such that -\pi \leq 0<\pi, \pi \leq 0<3 \pi, etc., the strips of Fig. 2-20 would be shifted vertically a distance \pi.
