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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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There is one more option in the stepping list that we haven t talked about yet, and that is Suspend . This option is interesting enough to merit its own section, and is discussed in section 9.4. In the meantime, let s finish our discussion of the SetPSDebug debugging features. 9.3.5 Catching undefined variables with strict mode The last debugging feature accessible through Set-PSDebug is strict mode .

This feature is conceptually similar to Option Explicit in Visual Basic or strict mode in PERL, and is named after the PERL feature, though it s not as rigorous as either the VB or PERL features. Still, it can make it easier to write a robust script.





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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

Normally in PowerShell, if a variable is undefined, it s treated as though it has the value $null. We can try this in an expression.

In this expression, $nosuchvariable is not defined. This means that it is treated as though it were $null. And $null in a numeric expression is treated as zero, so the whole expression evaluates to zero. It is important to note that the variable is treated as though it were null. This doesn t mean that there is now a variable called $nosuchvariable. We can verify this with the dir command:

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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

PS (2) > dir variable:\nosuchvariable Get-ChildItem : Cannot find path 'nosuchvariable' because it doe s not exist. At line:1 char:4 + dir <<<< variable:\nosuchvariable

and try the expression again:

PS (4) > 2 * $nosuchvariable The variable $nosuchvariable cannot be retrieved because it has not been set yet. At line:1 char:19 + 2 * $nosuchvariable <<<<

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

This time, you are sent an error telling you that the variable has not been defined. So let s define it.

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Run the expression again:

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

We get the expected result. Delete the variable:

PS (8) > 2 * $nosuchvariable The variable $nosuchvariable cannot be retrieved because it has not been set yet. At line:1 char:19 + 2 * $nosuchvariable <<<<

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

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