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1 y 1 y = 2 2 2 2 2 x + y 2 ( z + iy)( z iy) Therefore, there are two simple poles located at z = iy These lie directly on the y axis, one in the upper half plane and one in the lower half plane To get the right answer for the integral we seek, we need to compute using both cases First we consider the pole in the upper half plane The residue corresponding to z = + iy is a 1 = ( z iy) ye ikz 2 2 ( z + iy)( z iy) = yeky 1 ky e = 2 2 (2iy) 4 2i

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Now we can write our switch statement. The interesting clause is the one that handles -b. This clause uses the enumerator stored in $switch to advance the item 168

Applying the residue theorem, we nd that I= 1 2 y 1 1 e ky e ikx dx = 2 i 2 eky = x 2 + y2 4 i 2

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being processed to the next element in the list. We use a cast to [void] to discard the return value from the call to $switch.movenext() (more on that later). Then we use $switch.current to retrieve the next value and store it in $b. Then the loop continues processing the remaining arguments in the list.

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PS >> >> >> >> >> >> >> (4) > switch ($options) { '-a' { $a=$true } '-b' { [void] $switch.movenext(); $b= $switch.current } '-c' { $c=$true } '-d' { $d=$true } }

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However, now let s consider enclosing the other singularity, which would be an equally valid approach The singularity is located at z = iy, which is below the x axis, so we would need to use a semicircle in the lower half plane to enclose it This time the residue is ye ikz a 1 = ( z + iy) 2 2 ( z + iy)( z iy) Using this result, we obtain I= 1 2 y 1 1 e ky e ikx dx = 2 i 2 e ky = x 2 + y2 4 i 2

The last step in this example is to print out the arguments in the list to make sure they were all set properly.

We see that $a and $c are true, $b contains the argument Hello , and $d is still false since it wasn t in our list of test options. The option list has been processed correctly.

This isn t a robust example because it s missing all error handing. In a complete example, you would have a default clause that generated errors for unexpected options. Also, in the clause that processes the argument for -b , rather than discarding the result of MoveNext() it should check the result and generate an error if it returns false. This would indicate that there are no more elements in the collection, so -b would be missing its mandatory argument.

Combining both results gives the correct answer, which is I= EXAMPLE 711 Compute the integral given by I= SOLUTION This integral is given by I = 2 i residues in upper half plane We nd the residues by considering the complex function z2 f ( z) = 4 z +1 The singularities are found by solving the equation z4 + 1 = 0 This equation is solved by z = ( 1)1/ 4 But, remember that 1 = ei x2 dx 1 + x 4

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