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The result was an array of four elements 1,2,1,2 instead of 1,2,3. This is because the expression was parsed as (1,2,1)+2, building an array of three elements and then appending a fourth. You have to use parentheses to get the desired effect:

2 0

PS (2) > 1,2,(1+2) 1 2 3

The comma operator has higher precedence than any other operator except casts and property and array references. This is worth calling out again because it s important to keep in mind when writing expressions. If you don t remember this, you will produce some strange results.

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Use the result to deduce Poisson s integral formula for a circle of radius one: u (r , ) = 1 2

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The next thing to look at is nested arrays. Since a PowerShell array can hold any type of object, obviously it can also hold another array. We ve already mentioned that using the array subexpression operation was not the way to build a nested array. Now let s talk about how we actually do it using assignments and the comma operator. First, you can build nested arrays one piece at a time using assignments. Alternatively, you can just nest the comma operator within parentheses. Starting with last things first, here s how to build up a nested array structure using commas and parentheses. The result is concise:

PS (1) > $a = (((1,2),(3,4)),((5,6),(7,8)))

2 0

LISP users should feel fairly comfortable with this expression if they ignore the commas. Everybody else is probably shuddering.

And here s the same construction using intermediate variables and assignments. It s rather less concise but perhaps more easily understood.

$t1 = 1,2 $t2 = 3,4 $t3 = 5,6 $t4 = 7,8 $t1_1 = $t1,$t2 $t1_2 = $t3,$t4 $a = $t1_1, $t2_2

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SOLUTION We try separation of variables Let u (r , ) = R(r ) ( ) Then it follows that u R = ( ) r r 2u 2 R = ( ) r 2 r 2 2u 2 = R( r ) 2 2

In either case, what we ve done is built up a data structure that looks like the tree shown in figure 5.1: 124

Figure 5.1 A binary tree (arrays of arrays of arrays)

The statement of the problem tells us that 2u 1 u 1 2u =0 + + r 2 r r r 2 2 Hence 0= 2 2 R 1 R 1 ( ) + ( ) + 2 R(r ) 2 r r r r 2

The data structure in figure 5.1 is an array of two elements that are both arrays of two elements, which in turn contain arrays of two numbers.

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