Complex Analysis - Maths KU

190 Chapter 4 Elementary Functions
–4
–4
–3
–3
–2
4
3
2
1
y
–1 1 2 3 4
–1
–2
–3
–4
(a) The annulus 2≤|z|≤4
–2
4
3
2
1
v
–1 1 2 3 4
–1
–2
–3
–4
w = Ln z
(b) The image of the annulus in (a)
Figure 4.8 The mapping w =Lnz
x
u
EXAMPLE 6 Logarithmic Mapping
Find the image of the annulus2 ≤|z| ≤4 under the logarithmic mapping
w =Ln z.
Solution From property (ii) of logarithmic mapping, the boundary circles
|z| = 2 and |z| = 4 of the annulusmap onto the vertical line segments
u = log e 2 and u = log e 4, −π < v ≤ π, respectively. In a similar manner,
each circle |z| = r, 2 ≤ r ≤ 4, mapsonto a vertical line segment
u = log e r, −π