Complex Analysis - Maths KU

S 1
S 0
S –1
4π
3π
2π
π
– π
–2π
–3π
–4π
y
(a) Collection of half-infinite strips S n
v
(b) The image of each strip S n
is the punctured unit disk
Figure 4.9 The mapping w = e z
3π
2π
π
0
–π
–2π
–3π
–1
0
1 –1
Figure 4.10 A Riemann surface for
w = e z
0
x
u
1
4.1 Exponential and Logarithmic Functions 191
• Many propertiesof real logarithmsapply to the complex logarithm,
such as ln (z1z2) =lnz1 +lnz2, but these properties
don’t alwayshold for the principal value Ln z.
(ii) Since the complex exponential function isnot one-to-one, we can
use a Riemann surface, as described in the Remarks at the end
of Section 2.4, to help visualize the mapping w = e z . The Riemann
surface that we construct will also help us to visualize the
multiple-valued function w =lnz. Consider the mapping w = e z
on the half-plane x ≤ 0. Each half-infinite strip Sn defined by
(2n − 1)π