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.

Statement

Consider a complex-valued, continuous function, defined on a semicircular contour
of positive radius lying in the upper half-plane, centered at the origin. If the function is of the form
with a positive parameter, then Jordan's lemma states the following upper bound for the contour integral:
with equality when vanishes everywhere, in which case both sides are identically zero. An analogous statement for a semicircular contour in the lower half-plane holds when.

Remarks

Jordan's lemma yields a simple way to calculate the integral along the real axis of functions holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points,, …,. Consider the closed contour, which is the concatenation of the paths and shown in the picture. By definition,
Since on the variable is real, the second integral is real:
The left-hand side may be computed using the residue theorem to get, for all larger than the maximum of,, …,,
where denotes the residue of at the singularity. Hence, if satisfies condition, then taking the limit as tends to infinity, the contour integral over vanishes by Jordan's lemma and we get the value of the improper integral

Example

The function
satisfies the condition of Jordan's lemma with for all with. Note that, for,
hence holds. Since the only singularity of in the upper half plane is at, the above application yields
Since is a simple pole of and, we obtain
so that
This result exemplifies the way some integrals difficult to compute with classical methods are easily evaluated with the help of complex analysis.

Proof of Jordan's lemma

By definition of the complex line integral,
Now the inequality
yields
Using as defined in and the symmetry, we obtain
Since the graph of is concave on the interval, the graph of lies above the straight line connecting its endpoints, hence
for all, which further implies