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April 7th, 2013, 02:30 PM   #21
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Re: solutions to problems in Schaum's Complex Variables

The residue theorem for exterior domains states if is analytic outside (and on) a simple closed contour expect at isolated points (i.e., poles and isolated essential singularities), then where are the poles/isolated essential singularities outside of the contour and is the coefficient of the term in the Laurent expansion of that converges for greater than some .


The value is referred to as the residue at infinity.

So you can restate the theorem to say that .

The negative signs are there because if we're moving counterclockwise around the contour, then we're moving clockwise relative to the poles/isolated essential singularities outside of the contour.

You don't need to actually find the Laurent series since it turns out that calculating the residue at infinity is equivalent to calculating the residue of at the origin.

So


You don't have to expand to find that residue, of course. Since has a pole of order 4 at the origin and has a simple pole at the origin, has a pole of order at the origin.

And thus .


I'll try to think of an example where using the residue theorem for exterior domains to calculate a real-valued integral is much easier than using the regular residue theorem for interior domains. Or maybe you can think of an example.
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April 8th, 2013, 11:05 AM   #22
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Re: solutions to problems in Schaum's Complex Variables

Thanks G for the tutorial on the matter.
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