1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

480 CHAPTER 11 • APPLICATIONS OF H ARMONIC F UNCTIONS

y

v

2

t

w =Sl\V)

y!

v

Z = SJ<zl

W=SjZ)

Ro

-x

20 4-0

-20

'-40

Figure 11.62 The images of the circles Co and C1 under the composition mappings for
J (z) = S3 (S2 (S1z)).

Z, W, and w planes. First, the image of this circle Co under Z = 81 (z) is the

line Lo that passes through the origin and is inclined at the angle °'O· Second,

the function W = 82 ( Z) maps the line L 0 onto the ray Ro inclined at the angle

Zao. Finally, t he transformation given by w = 83 (W) maps the ray Ro onto the

arc of the circle Ao that passes through the points w2 = 2 and w 4 = - 2 and

intersects the ti-axis at w2 with angle 2ao, where ~ < Zao < 11'. T he restriction

on the angle ao, and hence 2a 0 , is necessary in order for the arc Ao to have a

low profile. The arc A 0 lies in the center of the Joukowski airfoil and is shown

in Figure 11.62.

If we let b be fixed, 0 < b < 1, then the larger circle C1 with center given

by c 1 = - h + i (1 + h) b will pass through the points z 2 = 1 and z 4 • = - 1 - Zh

and have radius r 1 = (1 + h) ./1 + b2. The circle C 1 also intersects the x-axis

at the point z2 at the angle a 0. The image of circle C 1 under Z = S 1 (z) is

u