 TOPICS  # 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