Jordan,s Lemma
المؤلف:
Arfken, G.
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press
الجزء والصفحة:
...
17-11-2018
925
Jordan's Lemma
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
and define the contour integral
 |
(6)
|
Then
Now, if 
, choose an 
such that 
, so
 |
(10)
|
But, for ![theta in [0,pi/2]](http://mathworld.wolfram.com/images/equations/JordansLemma/Inline25.gif)
,
 |
(11)
|
so
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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