# 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