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.