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When analyzing real integrals with contour integrals, how does one choose a proper contour integral?

Many cases can be solved by integrating around the top half of a circle with radius of infinity and then integrating along the entire real line.

I understand how when integrating one would avoid the branch cuts, but how would one know to use a rectangle or a quarter of a circle as a contour?

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    why this question has not received enough attention. its the same thing which I need to know more about it2013-09-11
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    The only guidance I've seen is that it is an art form... Doing/checking many examples helps.2014-04-21
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    @mhd.math It's the second thing that shows up on Google when I search "semicircle contour". I think it's getting plenty of attention ;)2017-03-01
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    @SimplyBeautifulArt ^_^ thank you2017-03-06

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