Jordan's lemma shows the value of the integral
(1)
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along the infinite upper semicircle and with is 0 for "nice" functions which satisfy . Thus, the integral along the real axis is just the sum of complex residues in the contour.
The lemma can be established using a contour integral that satisfies
(2)
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To derive the lemma, write
(3)
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(4)
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(5)
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and define the contour integral
(6)
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Then
(7)
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(8)
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(9)
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Now, if , choose an such that , so
(10)
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But, for ,
(11)
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so
(12)
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(13)
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(14)
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As long as , Jordan's lemma
(15)
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then follows.