1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10.3 • MAPPINGS INVOLVING ELEMENTARY FUNCTIONS 411

y

...... ..,.-........... -....... ~ ...... --· .... ,,

• in

w=exp z

z= -Log w

F igure 10.9 The conformal mapping w = exp z.

v

u

lwl < 1; the positive X-axis is mapped onto the lower semicircle; and the neg-
ative X -axis onto the upper semicircle. Figure 10.10 illustrates the composite
mapping.

• EXAMPLE 10.9 Show that the transformation w = f (z) = Log ( ~) is a

one-to-one conformal mapping of the unit disk lzl < 1 onto the horizontal strip
lvl < ~- Furthermore, the upper semicircle of the disk is mapped onto the line

v = ~ and the lower semicircle onto v = - 2 ".

Solution The function w = f (z) is the composition of the bilinear transfor-

mation Z = ~ followed by the logarithmic mapping w = Logz. The image

of the disk lzl < 1 under the bilinear mapping Z = ~~; is the right half-plane

Re(Z) > O; the upper semicircle is mapped onto the positive Y-axis; and the
lower semicircle is mapped onto the negative Y-axis. The logarithmic function

w = LogZ then maps the right half-plane onto the horizontal strip; the image

of the positive Y-axis is the line v = ~; and the image of the negative Y-axis is

the line v = - 2 ". Figure 10.11 shows the composite mapping.

• EXAMPLE 10.10 Show that the transformation w = f (z) = ( ~ )2 is a

one-to-one conformal mapping of the portion of the disk lzl < 1 that lies in the
upper half-plane Im(z) > 0 onto the upper half-plane Im (w) > 0. F\irthermore,
show that t he image of the semicircular portion of the boundary is mapped onto
the negative u-axis, and the segment - 1 < x < 1, y = 0 is mapped onto the
positive u-axis.