The Mapping 1/z

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Consider the transformation w = 1/z Notice that we can write w= 1 z y x iy x = 2 i 2 = = 2 x + y2 z zz ( x + iy)( x iy) x + y

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By inverting the transformation, it is easy to see that the coordinates in the z plane are related to coordinates in the w plane in the same way: x= Therefore the mapping w = 1/z Transforms lines in the z plane to lines in the w plane Transforms circles in the z plane to circles in the w plane u u + v2

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v u + v2

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Mapping and Transformations

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A circle that does not pass through the origin in the z plane is transformed into a circle not passing through the origin in the w plane A circle passing through the origin in the z plane is transformed into a line that does not pass through the origin in the w plane A line not passing through the origin in the z plane is transformed into a circle through the origin in the w plane A line through the origin in the z plane is transformed into a line through the origin in the w plane Let s try to understand how w = 1/z maps lines into lines A line in the complex plane that passes through the origin is a set of points of the form z = reia where a is some xed angle Under the mapping w = 1/z , we obtain a set of points: w= 1 1 ia = e z r

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This is another line that passes through the origin An important mapping w = 1/z transforms a disk in the z plane into the exterior of the disk in the w plane Consider the disk shown in Fig 910 The mapping w = 1/z maps this to the exterior of the circle of radius 1/r This is illustrated in Fig 911 The mapping w = 1/ z takes the point z = 0 to z = , and takes z = to z = 0

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Figure 910 A disk of radius r in the z plane

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Figure 911 The transformation w = 1/z has mapped the disk in Fig 99 to the entire w plane minus the region covered by the disk of radius 1/r

Mapping of In nite Strips

There are several important transformations that can be applied to in nite strips to map them to the upper half of the w plane Consider a strip of height a in the y direction that extends to along the x axis This is illustrated in Fig 912 When guring out how a transformation will work out, we pick out a few key points These are denoted by A-F in the gure The exponential function sends horizontal lines in the z plane into rays in the w plane That is, consider the transformation w = ez This maps the lines as shown in Fig 913

y C B a x D E F A

Figure 912 An in nite strip in the z plane

Mapping and Transformations

x w = ez

Figure 913 The exponential function maps horizontal lines to rays

If we apply the transformation w = e z / a (92)

to the in nite strip shown in Fig 912, the result is a mapping to the upper half of the w plane, shown in Fig 914 The points A, B, C, D, E, and F map to the points A , B , C , D , E , and F , respectively Now consider a vertical strip, as shown in Fig 915 We can map this to the upper half plane of Fig 914 using the transformation w = sin

z a

(93)

B 1

C D

E 1

F u

Figure 914 A mapping w = e z / a to the in nite strip shown in Fig 911 maps it to the upper half plane