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We have already seen that the modulus or magnitude or absolute value of a complex number is de ned by multiplying it by its complex conjugate and taking the positive square root The absolute value operator satis es several properties Let z1 , z2 , z3 , , zn be complex numbers Then z1z2 = z1 z2 z1z2 z3 zn = z1 z2 z3 zn z z1 = 1 z2 z2 A relationship called the triangle inequality deserves special attention: z1 + z2 z1 + z2 z1 + z2 + + zn z1 + z2 + z1 + z2 z1 z2 z1 z2 z1 z2 Also note that wz + zw = 2 Re( zw ) 2 z w + zn (127) (128) (129) (130) (124) (125)



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This script defines a function that prints out the information about where it was called from, and then calls the function it defined:

There are a number of interesting properties on the InvocationInfo object in $MyInvocation. We won t be covering them here, so take some time to explore it using Get-Member. The PowerShell software developer s kit also documents this object.





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We can figure out where the current function was called from. The trick is to find out where its parent was called from. This requires looking up the call stack and seeing what $MyInvocation was set to for the parent. The way to do that is to use the Get-Variable cmdlet. This cmdlet has a parameter that s used to specify what scope you want to get the variable from. If you specify scope 0, it looks the variable up in the current scope. If you specify 1 then the parent scope is searched, and so on. We ll take these two features and combine them in a function called gcs that will display the script call stack. Here s what that function looks like:

(126)

function { trap { 0..100 (gv } } gcs continue } | % { -scope $_ myinvocation).value.positionmessage -replace "`n"

1 Heydorn S, Johansen JD, Andersen KE, et al Fragrance allergy in patients with hand eczema a clinical study Contact Dermatitis 2003;48:317 2 Nielsen NH, Linneberg A, Menn T, et al Allergic contact sensitization in an adult Danish population: two cross-sectional surveys eight years apart (the Copenhagen Allergy Study) Acta Derm Venereol 2001;81:31

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This is a pretty simple function. It walks up the call stack, starting at 0 (current scope), until an error occurs. At that point, the trap statement catches the error and uses the continue statement to quietly exit the function. For each scope level, we print out a message displaying where it was called from. At this point we create a test script with a grouping of functions that call each other:

In Fig 11, we showed how a complex number can be represented by a vector in the x-y plane Using polar coordinates, we can develop an equivalent polar representation of a complex number We say that z = x + iy is the Cartesian representation of a complex number To write down the polar representation, we begin with the de nition of the polar coordinates (r , ): x = r cos y = r sin (131)

PS >> >> >> >> >> >> >> >> (1) > @' function a { b } function b { c } function c { d } function d { e } function e { gcs } a '@ > showstack.ps1

In this function, a calls b, which calls c, and so on until finally e calls our gcs function. Let s run it:

We have already seen that when we represent a complex number as a vector in the plane the length of that vector is r Hence, carrying forward with the vector analogy, the modulus of z is given by r = x 2 + y 2 = x + iy Using Eq (131), we can write z = x + iy as z = x + iy = r cos + ir sin = r (cos + i sin ) (132)

PS At At At (2) > ./showstack C:\Temp\showstack.ps1:5 char:17+ function e { gss <<<< } C:\Temp\showstack.ps1:4 char:15+ function d { e <<<< } C:\Temp\showstack.ps1:3 char:15+ function c { d <<<< }

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C:\Temp\showstack.ps1:2 char:15+ function b { c C:\Temp\showstack.ps1:1 char:15+ function a { b C:\Temp\showstack.ps1:6 char:2+ a <<<< line:1 char:11+ ./showstack <<<<

<<<< } <<<< }

(133)

We can see that the output starts with the innermost call and works its way back out until it reaches the line that called the script in the first place. Each line of output shows the file and line number where the function was called and what the line looks like. To make it even clearer, we ll modify the myinfo.ps1 script from earlier to call gcs and then call myinfo from showstack instead. (The actual modifications are left as an exercise for the reader.) Here s what the output of the revised set of scripts looks like:

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