Complex Analysis - Maths KU

440 Chapter 7 Conformal Mappings
y = π
Figure 7.58 Flow for Example 6
y = π
Figure 7.59 Flow for Example 7
y
y
x
x
Byreplacing the symbol z with the symbol w in the solution from Example
4, we obtain the mapping
z = Ω −1 (w) =w + Ln(w) + 1 (10)
of the upper half-plane v>0ontoD. The inverse Ω of the mapping in (10) is a
complex velocitypotential of a flow of an ideal fluid in D, but we cannot solve
for w to obtain an explicit formula for Ω. In order to describe the streamlines,
we recall that the streamlines in D are the images of horizontal lines v = c2
in the upper half-plane v>0 under the mapping z = w +Ln(w) + 1. Since
a horizontal line can be described by w(t) =t + ic2, −∞