1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

404 CHAPTER 10 • CONFORMAL MAPPING

y v

w=--i(I -Z)

l+z

Figure 10.5

i (1 - z)

The image of lzl < 1 under w =

1

.

+ z

onto in the extended complex plane, it follows that S maps the disk onto the

half-plane.

The general formula for a bilinear transformation (Equation (10-13)) appears

to involve four independent coefficients: a, b, c, d. But as S (z) f= I<, eit her a f= 0

or cf= 0, we can express the transformation with three unknown coefficients and

write either

z + k !!.!+!!

S (z) = --"- or S (z) = _c __ c

£!a +!! a z+!!' c

respectively. Doing so permits us to determine a unique bilinear transformation
if three distinct image values S(z1) = w1, S(z2) = w2, and S(z3) = W3 are
specified. To determine such a mapping, we can conveniently use an implicit
formula involving z and w.