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We call Eq (125) Poisson s formula for the half plane We can solve a wide variety of boundary value problems for a simply connected region R by using conformal mapping techniques The idea is to map the region R to the unit disk or to the half plane The mapping function that we use must be analytic There are three steps involved in obtaining a solution: Use a conformal transformation to map the boundary value problem for a region R to a boundary value problem on the unit disk or half plane Solve the problem using Eq (124) or (125) Find the inverse of the solution (that is apply the inverse of the conformal mapping) to write down the solution in the region R Remember that a simply connected region is one that includes no singularities While the mapping of the region R to a region R in the w plane must be conformal, the mapping of the boundary does not have to be conformal Three theorems are useful for solving these types of problems The rst tells us that a map w = f ( z ) has a unique inverse The second tells us that a harmonic function is transformed into a harmonic function under an analytic mapping w = f ( z ), and nally we learn that if the boundary value of a function is constant in the z plane, then it is in the w plane as well THEOREM 121 Let w = f ( z ) be analytic in a region R of the z plane Then there exists a unique inverse z = g( w) in R if f ( z ) 0 in R THEOREM 122 Let U ( x , y) be a function in some region R of the z plane and suppose that w = f ( z ) is analytic with f ( z ) 0 Then if (u, v ) is harmonic in the w plane where U ( z ) = ( w) are related by w = f ( z ), then U ( x , y) is harmonic THEOREM 123 Let U ( x , y) = A, where A is a constant on the boundary or part of the boundary C of a region R in the z plane Then its image (u, v ) = A on the boundary of the region R of the w plane If the normal derivative with respect to the boundary U / n = 0 in the z plane, then / n = 0 on the boundary of R in the w plane as well.

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PS (16) > $gd[13] = 3 Array assignment to [13] failed: The value "13" is not of type " System.String" and cannot be used in this generic collection. Parameter name: key. At line:1 char:5 + $gd[1 <<<< 3] = 3

Also note that this time, the PowerShell type converter didn t try to convert to the target type. The type signature is too complex for it to figure out, so we ll have to explicitly cast the arguments. Finally, let s look at the full name of the type of object stored in $gd.

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Notice that moduli multiply and arguments add Division is also very simple: z rei r i i r i( ) = = e e = e w ei The reciprocal of a complex number takes on the relatively simple form: 1 z = rei z 1 = e i r Raising a complex number to a power is also easy: z n = (rei )n = r n ein The complex conjugate is just z = re i (146) (145) (144) (143)

PS (17) > $gd.gettype().FullName System.Collections.Generic.Dictionary`2[[System.String, mscorlib , Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e 089],[System.Int32, mscorlib, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e089]] PS (18) >

To reiterate the comment from the beginning of this section, very rarely will you need some of these esoteric capabilities, but when you do need them, they can be incredibly valuable. Enough with the .NET trivia challenge. In the remainder of this chapter, we ll look at how we can apply some of the things we ve learned to build more interesting applications.

Euler s formula can be used to derive some interesting expressions For example, we can easily derive one of the most mysterious equations in all of mathematics: ei = cos + i sin ei + 1 = 0 (147)

These examples are designed to demonstrate the range of tasks that can be addressed with PowerShell rather than be strictly practical. For some more purely practical examples using .NET in admin scenarios, take a look at appendix B.

In this section, we re going to put the network back into .NET. As one would expect from a modern programming environment, .NET (and consequently PowerShell) has a pretty comprehensive set of types for doing network programming.

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