Suppose the circle C of Problem 8.27 is moved so that its center is in the upper half plane but that it still passes through z = 1 and encloses z = -1. Determine the image of C under the transformation w = 1/2 (z + 1/z).

Question

Suppose the circle [latex]C[/latex] of Problem 8.27 is moved so that its center is in the upper half plane but that it still passes through [latex]z=1[/latex] and encloses [latex]z=-1[/latex]. Determine the image of [latex]C[/latex] under the transformation [latex]w=\frac{1}{2}(z+1 / z)[/latex].

Accepted Answer

As in Problem 8.27, since [latex]z=1[/latex] is a critical point, we will obtain the sharp tail at [latex]w=1[/latex] [Fig. 8-90]. If [latex]C[/latex] does not entirely enclose the circle [latex]|z|=1[/latex] [as shown in Fig. 8-89], the image [latex]C^{\prime}[/latex] will not entirely enclose the image of [latex]|z|=1[/latex] [which is the slit from [latex]w=-1[/latex] to [latex]w=1[/latex] ]. Instead, [latex]C^{\prime}[/latex] will only enclose that portion of the slit which corresponds to the part of [latex]|z|=1[/latex] inside [latex]C[/latex]. The appearance of [latex]C^{\prime}[/latex] is therefore as shown in Fig. 8-90. By changing [latex]C[/latex] appropriately, other shapes similar to [latex]C^{\prime}[/latex] can be obtained.

The fact that [latex]C^{\prime}[/latex] resembles the cross-section of the wing of an airplane, sometimes called an airfoil, is important in aerodynamic theory (see Chapter 9) and was first used by Joukowski. For this reason, shapes such as [latex]C^{\prime}[/latex] are called Joukowski airfoils or profiles and [latex]w=\frac{1}{2}(z+1 / z)[/latex] is called a Joukowski transformation.

Question 8.28: Suppose the circle C of Problem 8.27 is moved so that its ce...

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