1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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11.8 • THE JOUKOWSKI AlRFOIL 479

w = J(z)

IV = Sz(ZJ V l w=S)/'W)

Figure 11.61 The composition mappings for J (z) = 83 (82 (Si (z))).

Z = S1 (z) maps the region lzl > 1 onto the right half-plane Re(Z ) > 0, and

the points z 1 = -i, z2 = 1, zs = i, and Z4 = - 1 are mapped onto Z1 = -i,

Z2 = 0, Z3 = i, and Z 4 = ioo, respectively. Second, the function W = 82 (Z)

maps the right half-plane onto the W plane slit along its negative real axis,
and the points Z 1 = -i, Z2 = 0, Za = i, and Z4 = ioo are mapped onto

W1 = - 1, W 2 = 0, W 3 = - 1, and W 4 = -oo, respectively. T hen the bilinear

transformation w = S 3 (W) maps the latter region onto the w plane slit along

the portion of the real axis -2 ::; u ::; 2, and the points W1 = - 1, W2 = 0,

Ws = - 1, and W 4 = -oo are mapped onto w1 = 0 , W2 = 2, Ws = 0, and

W4 = - 2, respectively. These three compositions are shown in Figure 11.61.

The circle Co with center co = ia on the imaginary axis passes through the

points z 2 = 1 and z 4 = -1 and has radius r 0 = ~. With the restriction

that 0 < a < 1, then this circle intersects the x-axis at the point z2 with angle

o 0 = ~ - Arctan a, with ~ < oo < ~. We want to track the image of Co in the