Jordan lemma

Let $ f( z) $ be a regular analytic function of a complex variable $ z $, where $ | z | > c \geq 0 $, $ \mathop{\rm Im} z \geq 0 $, up to a discrete set of singular points. If there is a sequence of semi-circles

$$ \gamma ( R _ {n} ) = \{ {z } : {| z | = R _ {n} ,\ \mathop{\rm Im} z \geq 0 } \} ,\ R _ {n} \uparrow + \infty , $$

such that the maximum $ M ( R _ {n} ) = \max | f ( z) | $ on $ \gamma ( R _ {n} ) $ tends to zero as $ n \rightarrow \infty $, then

$$ \lim\limits _ {n \rightarrow \infty } \int\limits _ {\gamma ( R _ {n} ) } e ^ {iaz} f ( z) dz = 0 , $$

where $ a $ is any positive number. Jordan's lemma can be applied to residues not only under the condition $ zf ( z) \rightarrow 0 $, but even when $ f ( z) \rightarrow 0 $ uniformly on a sequence of semi-circles in the upper or lower half-plane. For example, in order to calculate integrals of the form

$$ \int\limits _ {- \infty } ^ \infty f ( x) \cos a x \ d x ,\ \int\limits _ {- \infty } ^ \infty f ( x) \sin a x d x . $$

Obtained by C. Jordan [1].

References

[1]	C. Jordan, "Cours d'analyse" , 2 , Gauthier-Villars (1894) pp. 285–286
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1967) (In Russian)
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6

References

[a1]	D.S. Mitrinović, J.D. Kečkić, "The Cauchy method of residues: theory and applications" , Reidel (1984)