1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 215

y

W=z+b

(^0) x
Fig. 5.1
transformed into straight lines, and circles into congruent circles. Angles
between arcs will be preserved in magnitude and orientation.
5.2 THE SIMILITUDE w = az (a ~ 0)
In this case we have
lwl = lallzl
and
arg w = Arg a + Arg z
Hence the mapping defined by w = az consists of an homotecy (or dilation)
with center 0 and ratio lal, followed by a rotation of amplitude a = Arg a,
or vice versa (Fig. 5.2). Consequently, each geometric figure in the plane
is mapped by this function into a directly similar figure. In particular,
straight lines are mapped into straight lines and circles into circles. Angles
between arcs are preserved in magnitude and orientation.
Y w = az

'

0

Fig. 5.2

'

' '

a

lalz

x