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java barcode scanner api In Example 94, we start with a region de ned by a disk centered at z = 1 + i in Java
In Example 94, we start with a region de ned by a disk centered at z = 1 + i QR Code 2d Barcode Drawer In Java Using Barcode generation for Java Control to generate, create QR Code JIS X 0510 image in Java applications. QR Code 2d Barcode Scanner In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Complex Variables Demysti ed
Creating Barcode In Java Using Barcode generator for Java Control to generate, create barcode image in Java applications. Barcode Recognizer In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Figure 921 A disk at the origin is obtained from the disk shown in Fig 920 via the transformation Z = z z0 = z + 1 i We denote this the Z plane Encoding QR Code In C# Using Barcode generator for VS .NET Control to generate, create QRCode image in Visual Studio .NET applications. QR Code Printer In Visual Studio .NET Using Barcode creator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. The rst step is to move the disk to the origin We do this using Z = z z0 = z + 1 i The result is the disk shown in Fig 921 Now we want to transform the disk shown in Fig 921 so that the region of de nition is the entire complex plane minus a hole where the disk was We do this using an inverse transformation: w= The result is shown in Fig 922 1 Z QR Creator In VS .NET Using Barcode drawer for .NET framework Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. Encode QR Code 2d Barcode In Visual Basic .NET Using Barcode generator for .NET Control to generate, create QR Code image in .NET applications. Figure 922 The transformation 1/Z changes the region to the entire complex plane with a hole punched out in the middle The radius of the hole is 1/r if the radius of the disk we started with was Z = rei In our example, r = 2 so the hole here has a radius = 1/ 2 Barcode Generator In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. Creating Code39 In Java Using Barcode generation for Java Control to generate, create Code 3/9 image in Java applications. Mapping and Transformations
Code128 Drawer In Java Using Barcode printer for Java Control to generate, create Code 128A image in Java applications. Data Matrix Generation In Java Using Barcode encoder for Java Control to generate, create Data Matrix 2d barcode image in Java applications. The complete transformation in this example can be written as w= 1 z +1 i
ISSN Generator In Java Using Barcode maker for Java Control to generate, create International Standard Serial Number image in Java applications. Painting USS Code 39 In Java Using Barcode creation for Eclipse BIRT Control to generate, create Code39 image in BIRT reports applications. This is a M bius transformation as in Eq (95) with a = 0, b = 1, c = 1, and d = 1 + i EXAMPLE 95 Construct a M bius transformation that maps the unit disk to the left half plane Re( z ) < 0 and one that maps the unit disk to the right half plane Re( z ) > 0 SOLUTION The rst transformation we want to consider is illustrated in Fig 923 First we consider the boundary of the disk, which is the unit circle, that is the set of points  z  = 1 For the transformation shown in Fig 923 to work, we must map the points on the unit circle to the imaginary axis In the form of a M bius transformation, the mapping will be of the form Tz = az + b cz + d Code39 Decoder In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Drawing USS Code 128 In .NET Using Barcode printer for Visual Studio .NET Control to generate, create Code128 image in .NET applications. This transformation has a pole located at the point z = d /c We are free to pick a point on the unit circle to map to the pole, so we choose z = 1 With this choice we have the freedom to x c and d, so we choose c = 1, d = 1 So Tz = az + b z 1 Bar Code Scanner In .NET Framework Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Barcode Generation In None Using Barcode generation for Office Excel Control to generate, create bar code image in Office Excel applications. Figure 923 We want to map the unit disk to the left half plane
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Second, we need to pick a point z on the unit circle such that Tz = 0 We have already used the point z = 1, so we choose z = 1 This forces us to take a = b since Tz = 0 when z = 1 We can choose a = 1 giving us the transformation Tz = z +1 z 1 Now we can see how the transformation maps the rest of the unit disk The simplest point to check is the point z = 0 at the center of the disk We have Tz(0) = 0 +1 = 1 0 1 So the transformation maps the center of the disk z = 0 to the point z = 1 which is in the left half plane Hence this is the transformation that we want To transform the unit disk to the right half plane instead, it turns out we are almost there All we have to do is rotate the transformed region shown in Fig 922 The angle that is required is , and a rotation is implemented by multiplication by ei So the transformation that takes the unit disk to the right half plane is given by Tz = ei z +1 z + 1 = z 1 z 1 EXAMPLE 96 Consider a mapping that will transform the unit disk into the upper half plane
SOLUTION We again seek a transformation of the form Tz = az + b cz + d
This time we choose to map the point z = 1 onto the pole at z = d /c If we choose c = 1, then d = 1 as well and the transformation is given by Tz = az + b z +1

