Jordan's lemma

In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan. lim R → ∞ M R = 0 {displaystyle lim _{R o infty }M_{R}=0}     (*) In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan. Consider a complex-valued, continuous function f, defined on a semicircular contour of positive radius R lying in the upper half-plane, centered at the origin. If the function f is of the form with a positive parameter a, then Jordan's lemma states the following upper bound for the contour integral: where equal sign is when g vanishes everywhere. An analogous statement for a semicircular contour in the lower half-plane holds when a < 0. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = ei a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z1, z2, …, zn. Consider the closed contour C, which is the concatenation of the paths C1 and C2 shown in the picture. By definition, Since on C2 the variable z is real, the second integral is real: The left-hand side may be computed using the residue theorem to get, for all R larger than the maximum of |z1|, |z2|, …, |zn|, where Res(f, zk) denotes the residue of f at the singularity zk. Hence, if f satisfies condition (*), then taking the limit as R tends to infinity, the contour integral over C1 vanishes by Jordan's lemma and we get the value of the improper integral