Complex Analysis - Maths KU

∇ φ = 0
2
φ = k1 ∇ φ = 0
2
y
φ = k0 dφ
= 0
dn —
φ
d
dn —
a dφ
= 0 b
dn —
Figure 7.83 Figure for Problem 23
x
7.5 Applications 445
15. y
16.
π i
0
Figure 7.77 Figure for Problem 15
x
i
y
1
Figure 7.78 Figure for Problem 16
In Problems 17–20, the flow of an ideal fluid is shown in a domain in the z-plane.
(a) Find a conformal mapping of the upper half-plane w>0 onto the domain in
the z-plane, and (b) find a parametric representation of the streamlines of the flow.
17. y = π y
18.
y = π/2
Figure 7.79 Figure for Problem 17
19. y
20.
i
Figure 7.81 Figure for Problem 19
x
x
Figure 7.80 Figure for Problem 18
i
Figure 7.82 Figure for Problem 20
In Problems 21 and 22, construct the flow of an ideal fluid in the given domain with
sinks or sources on the boundary of the domain.
21. The domain from Problem 13 with a source at z1 =1+i and a sink at z2 =2
22. The domain from Problem 16 with a source at z1 = 1
2
z2 = 2 and z3 =3i
Focus on Concepts
y
i
y
√ √
1
2+ 2i and sinks at
2
23. Show that the function given by (3) with the symbols u and v replaced by the
symbols x and y is a solution of the boundary-value problem in the domain
shown in color in Figure 7.83.
x
x
x