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By computing the Laurent expansion of an analytic function in a region containing one or more singularities, we were able to arrive at the residue theorem which can be used to calculate a wide variety of integrals This includes integrals of complex functions, but the residue theorem can also be used to calculate certain classes of integrals involving functions of a real variable



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This shows that we can get an arbitrary property out of an object, and then do arbitrary processing on that information using the Foreach-Object command. Combining those features, here s an example that adds up the lengths of all of the objects in a directory.

PS (3) > $a[1] 2 3

3 Find the principal part of f ( z ) =

However, if we simply run it through Foreach-Object, we ll find that the length of the result is now three, and the second element in the result is simply the number 2 . 172 FLOW CONTROL IN SCRIPTS

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PS (4) > $b = $a | foreach { $_ } PS (5) > $b.length 3 PS (6) > $b[2] 2

In effect, the result has been flattened . However, if we use the unary comma operator before the $_ variable, the result has the same structure as the original array.

1 (1 + z 3 )2

PS (7) > $b = $a | foreach { , $_ } PS (8) > $b.length 2 PS (9) > $b[1] 2 3

When chaining foreach cmdlets, we need to repeat the pattern at each stage:

sin z 5 z z + 2 sin z 5 What are the singularities and residues of 2 z ( z ) d 6 Evaluate 2 0 24 6 sin 4 What are the singular points and residues of

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Green tea has been consumed as a popular beverage in Asian countries, in particular, and throughout the world for many years It has recently gained greater popularity in western nations, though, because of its purported antioxidant and anticarcinogenic effects There are numerous in vitro and in vivo studies on the effects of green tea on the skin; in fact, green tea is one of the most studied antioxidants133 Besides its antioxidant activity, green tea polyphenols possess anti-inflammatory and anticarcinogenic properties and modulate the biochemical pathways important in cell proliferation, when administered either topically or orally134,135 Polyphenols are a large, diverse family of thousands of chemical substances found in plants Significantly, many of them are strong antioxidants (Table 34-1) There are four major polyphenolic catechins in green tea: ECG [(-)EpiCatechin3-O-Gallate], GCG [(-)GalloCatechin-3O-Gallate], EGCG [(-)EpiGalloCatechin -3-O-Gallate], and EGC [(-)EpiGallo Catechin] EGCG is the most abundant and biologically active component (Fig 34-5) Special preparation of the tea plant ensures that the antioxidant activities of these polyphenols are preserved White tea, like green tea, comes from the Camellia sinensis plant However, white tea is more expensive because it is harder to obtain White tea actually comes from the tips of the green tea leaves or from leaves that are not yet fully open and the buds are still covered by fine white hair EGCG is the main compound in green and white tea responsible for antioxidant activity136,137 When evaluating the utility of a skin care product, it is necessary to know which polyphenols are in the product EGCG is the most potent form of green tea polyphenols In addition, it is necessary to know what percent of polyphenols

PS (7) > $b = $a | foreach { , $_ } | foreach { , $_ } PS (8) > $b.length 2 PS (9) > $b[1] 2 3

Why did we do this Why didn t we just preserve the structure as we pass the elements through instead of unraveling by default Well, both behaviors are, in fact, useful. Consider the follow example, which returns a list of loaded module names:

dx 8 Use the residue theorem to compute lim R R 1 + x 2 x2 + 3 9 Compute ( x 2 + 1)( x 2 + 4) dx cos x 10 Compute dx 1 + x 2

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