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The next step in extending the calculus of real variables to include complex variables is to de ne the notion of a derivative You won t be surprised to nd out that computing derivatives or shall we say determining when a function is differentiable is a little more involved when considering functions of a complex variable While we will see that many of the basic theorems about derivatives carry over from real to complex variables, there are some differences In particular, we re going to have to pay attention to how we approach the origin when computing the limits used to de ne the derivative, and we ll nd the unusual result that some functions of a complex variable are continuous but not differentiable After learning this we ll see that functions of a complex variable that are differentiable satisfy a nice set of equations known as the Cauchy-Riemann equations, one of the most elegant results in pure and applied mathematics Loosely speaking, a function which satis es the Cauchy-Riemann equations is



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Now let s get the same information from the registry:

PS (8) > dir hklm:\software | get-member ps* TypeName: Microsoft.Win32.RegistryKey Name ---PSChildName PSDrive PSIsContainer PSParentPath PSPath PSProvider MemberType ---------NoteProperty NoteProperty NoteProperty NoteProperty NoteProperty NoteProperty Definition ---------System.String PSChildName=Adobe System.Management.Automation.PSDri... System.Boolean PSIsContainer=True System.String PSParentPath=Microso... System.String PSPath=Microsoft.Pow... System.Management.Automation.Provi...





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You can see the same set of PS* properties with the PowerShell (PS) prefix on the object, even though they are completely different types. Take a look at these properties. They allow you to work with these two different objects in the same way. This means that you can always tell if an object might have children by looking at the PSIsContainer property, regardless of the type of the underlying object. And you can always get the path to the object through the PSPath property. We call this type of adaptation object normalization. By adding this set of synthetic properties to all objects returned from the provider infrastructure, it becomes possible to write scripts that are independent of the type of object that the provider surfaces. This makes the scripts both simpler and more reusable. 8.2.3 Using Add-Member to extend objects The Add-Member cmdlet is the easiest way to add a new member to an object instance, either an existing object or a synthetic object. It can be used to add any type of member supported by the PowerShell type system. The list of possible member types that can be added with Add-Member is shown in table 8.1. We ll work through some examples showing how to use these members. We ll use an instance of the string Hi there to do this. For convenience, we ll store it in a variable $s as shown:

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called analytic Finally, we close the chapter with a look at a special type of function that satis es Laplace s equation which we call a harmonic function Harmonic functions are functions of the real variables x and y but they can be used to construct a complex function, which is analytic

PS (1) > $s = "Hi there"

solution of toluidine blue and basic fuchsin in 30% alcohol, followed by rinsing with regular tap water Measurements of the colors of the sample are made using a reflectance colorimeter Then, a trained person scores the discs with a microscope at 20x magnification as follows: Intercorneocyte cohesion: 0 large sheet; 1 large clusters a few isolated cells; 2 small clusters many isolated cells; 3 large amount of dye in cells, but uniform; 4 important staining in all cells, often with grains20

The Match() operator returns a Match object (full name System.Text.RegularExpressions.Match). We can use the Get-Member cmdlet to explore the full set of members on this object at our leisure, but for now we re interested in only three members. The first member is the Success property. This will be true if the pattern matched. The second interesting member is the Value member, which will contain the matched value. The final member we re interested in is the NextMatch() method. Calling this method will step the regular expression engine to the next match in the string, and is the key to tokenizing an entire expression. We can use this method in a while loop to extract the tokens from the source string one at a time. In the example, we keep looping as long the Match object s Success property is true. Then we display the Value property and call NextMatch() to step to the next token:

Consider some point z0 in the complex plane and let f ( z ) be some function such that its domain contains a neighborhood of z0 The derivative of f ( z ) at the point z0 is de ned by the limit f ( z0 ) = lim

PS (6) > while ($m.Success) >> { >> $m.value >> $m = $m.NextMatch() >> } >> 11 + 2 * 35 4

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