2.5 • THE RECIPROCAL TRANSFORMATION w = ~ 85(a) Show that f is, in general, an n -valued function.
(b) Write the principal nth root function.
(c) Write a branch of t he multivalued nth root function that is d ifferent
from the one in part (b).- Describe a Riemann sur face for the domain of definition of the multivalued function
1
(a) w = f (z) = z•.
(b) w=f(z)=zt.
10. Discuss how Riemann surfaces should be used for both the domain and the range
2
to help describe the behavior of the multivalued function w = f (z) = z~.2.5 THE RECIPROCAL TRANSFORMATION
w =!
zThe mapping w = f (z) = ~ is called the reciprocal transformation and maps
the z plane one-to-one and onto t he w plane except for the point z = O, which
has no image, and the point w = 0, which has no preimage or inverse image.
Using exponential notation w = pei, if z = rei^8 f= 0, we have
W=pe"'=i'-^1 - =^1 - e- i O.
z r
(2-31)The geometric description of the reciprocal transformation is now evident.
It is an inversion (that is, the modulus of ~ is t he reciprocal of the modulus of
z) followed by a reflection through the x-axis. The ray r > 0, 0 =a, is mapped
one-to-one and onto the ray p > 0, </! = - a. Points that lie inside the unit circle
C 1 (O) = {z: lzl = 1} are mapped onto points that lie outside the unit circle,
and vice versa. The situation is illustrated in Figure 2.21.
yw=!
'Figure 2.2 1 The reciprocal transformation w = ~.
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