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Date: 11-12-2018
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Date: 18-10-2018
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Jordan's lemma shows the value of the integral
(1) |
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) |
To derive the lemma, write
(3) |
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(4) |
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(5) |
and define the contour integral
(6) |
Then
(7) |
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(8) |
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(9) |
Now, if , choose an such that , so
(10) |
But, for ,
(11) |
so
(12) |
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(13) |
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(14) |
As long as , Jordan's lemma
(15) |
then follows.
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 406-408, 1985.
Jordan, C. Cours d'Analyse de l'Ecole polytechnique, Tome 2, 3. éd., rev. et corrigé. Paris: Gauthier-Villars, pp. 285-86, 1909-1915.
Whittaker, E. T. and Watson, G. N. "Jordan's Lemma." §6.222 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 115-117, 1990.
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