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Complex Variables Demysti ed
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Figure 915 To map a vertical strip to the upper half plane, we utilized the transformation w = sin z / a It maps to the region shown in Fig 916
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Here are a few basic rules of thumb to consider when doing transformations They can be illustrated considering the square region shown in Fig 96 In that section, we saw how to shift the position of the square by adding a constant, that is, we wrote down a linear transformation of the form w = z + a Let s review the other types of transformations that are possible A transformation of the type w = az will expand the region if a > 1 and will shrink the region if a < 1 Consider the rst case with w = 2 z This expands the square region from 0 x 1, 0 y 1 to 0 x 2, 0 y 2 This is illustrated in Fig 917
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The transformation w = sin( z / a) maps the region in Fig 915 to the region in Fig 916 with corresponding points indicated
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Mapping and Transformations
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w = 2z
Figure 917 For review, a transformation of the form w = az expands the region of interest if | a | >1
Now suppose that w = (1/ 2) z This shrinks the square, as shown in Fig 918 To rotate the region by an angle , we use a transformation of the form w = ei z (94)
For our square, this rotates the square by in the counterclockwise direction assuming that > 0 This is illustrated in Fig 919
M bius Transformations
In this section, we consider a transformation of the type: Tz = az + b cz + d ad bc 0 (95)
w = 1/2z
Figure 918 We shrink the square by the transformation w = az when | a | <1
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w = eifz f
Figure 919 A rotation is implemented with a transformation of the form w = ei z
This type of transformation goes under the various names bilinear transformation, fractional transformation, or M bius transformation The transformation shown in Eq (95) is actually a composition of three different transformations These are Dilation, which can be written as the linear transformation az Translation, which is written as z + b Reciprocation, which is the transformation 1/z The requirement that ad bc 0 is based on the following The derivative of Eq (95) is given by (Tz ) = a(cz + d ) c(az + b) (cz + d )2
Evaluating this at z = 0 we have (Tz ) (0) = ad bc (d ) 2
This tells us that the transformation in Eq (95) will be a constant unless ad bc 0 A transformation of the type in Eq (95) maps circles in the z plane to circles in the w plane Straight lines are also mapped into straight lines Now suppose that z0 , z1 , z 2 , and z3 are four distinct points in the complex plane The cross ratio is given by ( z3 z0 )( z1 z 2 ) ( z1 z0 )( z3 z2 )
(96)
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The cross ratio is invariant under a M bius transformation That is if z j w j under a M bius transformation, then ( z3 z0 )( z1 z 2 ) ( w w0 )( w1 w2 ) 3 ( z1 z0 )( z3 z2 ) ( w1 w0 )( w3 w2 ) There are a few M bius transformations of interest Let a be a complex number with | a | < 1 and suppose that | k | = 1 Then w=k z a 1 az (97)
maps the unit disk from the z plane to the unit disk in the w plane Now let a be a complex number with the requirement that Im(a) > 0 The transformation w=k z a z a (98)
maps the upper half of the z plane to the unit disk in the w plane Notice that when z is purely real, | w | = | k | = 1 EXAMPLE 94 Consider a disk of radius r = 2 centered at the point z = 1 + i Find a transformation that will take this to the entire complex plane with a hole of radius 1/2 centered at the origin SOLUTION Since these transformations are linear, we can do this by taking multiple transformations in succession First we illustrate what we re starting with, a disk of radius r = 2 centered at the point z = 1 i This is shown in Fig 920
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