.1 Center: Radius: .5
Center: .2 + i Radius: 1
Center: 1 + i Radius: 1
Center: -2 + 3 i Radius: 3 2 4.2426
Center: .2 + i Radius: 1.2806 Figure 7.21.
Center: .1 + .3 i Radius: .9487
Center: .1 + .1 i Radius: 1.1045
Center: -.2 + .1 i Radius: 1.2042
Airfoils Obtained from Circles via the Joukowski Map.
rest of the plane, as do the images of the (nonzero) points inside the unit circle. Indeed, if we solve (7.67) for (7.68) z = 2 - 1 , we see that every except 1 comes from two different points z; for not on the critical line segment [ - 1, 1 ], one point lies inside and and one lies outside the unit circle, whereas if -1 < < 1, the points lie on the unit circle and on a common vertical line. Therefore, (7.67) defines a one-to-one conformal map from the exterior of the unit circle | z | > 1 onto the exterior of the unit line segment C \ [ -1, 1 ]. Under the Joukowski map, the concentric circles | z | = r = 1 are mapped to ellipses with foci at 1 in the plane; see Figure 7.20. The effect on circles not centered at the origin is quite interesting. The image curves take on a wide variety of shapes; several examples are plotted in Figure 7.21. If the circle passes through the singular point z = 1, then its image is no longer smooth, but has a cusp at = 1; this happens in the last 6 of the figures. Some of the image curves have the shape of the cross-section through an airplane wing or airfoil. Later, we will see how to construct the physical fluid flow around such an airfoil, a result that was a critical step in early aircraft design. Composition and the Riemann Mapping Theorem One of the features of conformal mapping is that one can assemble a large repertoire of complicated examples by simply composing elementary mappings. The method rests on the simple fact that the composition of two complex analytic functions is also complex analytic. This is the complex counterpart of the result, learned in first year calculus, that the composition of two differentiable functions is itself differentiable. 1/19/12 249
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w = ez
=
w-1 w+1
Figure 7.22.
Composition of Conformal Maps.
Proposition 7.31. If w = f (z) is an analytic function of the complex variable z = x + i y, and = g(w) is an analytic function of the complex variable w = u + i v, then the composition = h(z) g f (z) = g(f (z)) is an analytic function of z. The proof that the composition of two differentiable functions is differentiable is identical to the real variable version, [7, 129], and need not be reproduced here. The derivative of the composition is explicitly given by the usual chain rule: d g f (z) = g (f (z)) f (z), dz or, in Leibnizian notation, d dw d = . dz dw dz (7.69)
If both f and g are one-to-one, so is their composition h = g f . Moreover, the composition of two conformal maps is also conformal, a fact that is immediate from the definition, or by using the chain rule (7.69) to show that h (z) = g (f (z)) f (z) = 0 provided g (f (z)) = 0 and f (z) = 0.
Example 7.32. As we learned in Example 7.22, the exponential function w = ez
1 maps the horizontal strip S = { - 1 < Im z < 2 } conformally onto the right half plane 2 R = { Re w > 0 }. On the other hand, Example 7.24 tells us that the linear fractional w-1 transformation = w+1
maps the right half plane R conformally to the unit disk D = { | | < 1 }. Therefore, the composition ez - 1 (7.70) = z e +1 is a one-to-one conformal map from the horizontal strip S to the unit disk D, which we illustrate in Figure 7.22.
Of course, to properly define the composition, we need to ensure that the range of the function w = f (z) is contained in the domain of the function = g(w).
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Recall that our motivating goal is to use analytic functions/conformal maps to solve boundary value problems for the Laplace equation on a complicated domain by transforming them to boundary value problems on the unit disk. Of course, the key question the student should be asking at this point is: Is there, in fact, a conformal map = g(z) from a given domain to the unit disk D = g()? The theoretical answer is the celebrated Riemann Mapping Theorem. Theorem 7.33. If C is any simply connected open subset, not equal to the entire complex plane, then there exists a one-to-one complex analytic map = g(z), satisfying the conformality condition g (z) = 0 for all z , that maps to the unit disk D = { | | < 1 }. Thus, any simply connected domain -- with one exception, the entire complex plane -- can be conformally mapped the unit disk. Note that need not be bounded for this to hold. Indeed, the conformal map (7.57) takes the unbounded right half plane R = { Re z > 0 } to the unit disk. The proof of this important theorem relies on some more advanced results in complex analysis, and can be found, for instance, in [3]. The Riemann Mapping Theorem guarantees the existence of a conformal map from any simply connected domain to the unit disk, but its proof is not constructive, and so is of little help for constructing the desired mapping. And, in general, this is not an easy task. In practice, one assembles a collection of useful conformal maps that apply to particular domains of interest. An extensive catalog can be found in [74]. More complicated maps can then be built up by composition of the basic examples. Ultimately, though, the determination of a suitable conformal map is more an art than a systematic science. Example 7.34. Suppose we are asked to conformally map the upper half plane U = Im z > 0 to the unit disk D = | | < 1 . We already know that the linear fractional transformation w-1 = g(w) = w+1 maps the right half plane R = Re w > 0 to D = g(R). On the other hand, multiplication by i = e i /2 , with z = h(w) = i w, rotates the complex plane by 90 and so maps the right half plane R to the upper half plane U = h(R). Its inverse h-1 (z) = - i z will therefore map U to R = h-1 (U ). Therefore, to map the upper half plane to the unit disk, we compose these two maps, leading to the conformal map = g h-1 (z) = iz + 1 -iz - 1 = -iz + 1 iz -1 (7.71)
from U to D. In a similar vein, we already know that the squaring map w = z 2 maps the upper 1 right quadrant Q = 0 < ph z < 2 to the upper half plane U . Composing this with our previously constructed map -- which requires replacing z by w in (7.71) beforehand -- leads to the conformal map i z2 + 1 (7.72) = i z2 - 1 that maps the quadrant Q to the unit disk D. 1/19/12 251
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