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# 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

 (3) (4) (5)

and define the contour integral

 (6)

Then

 (7) (8) (9)

Now, if , choose an such that , so

 (10)

But, for ,

 (11)

so

 (12) (13) (14)

As long as , Jordan's lemma

 (15)

then follows.

Contour Integration

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## 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.

Jordan's Lemma

## Cite this as:

Weisstein, Eric W. "Jordan's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JordansLemma.html