Jordan's lemma

The lemma of Jordan (after Marie Ennemond Camille Jordan ) is an auxiliary function theory. It is used together with the residue theorem to calculate integrals of real analysis.

Statement

Is and converges in the upper half-plane uniformly to zero for all, then applies

For.

This is true even if is in addition uniformly pursued in the upper half-plane zero. Entirely analogously can formulate the lemma for the lower half-plane.

Application

Many of improper integrals of the form can be, if they exist, calculated in the following manner: integrated on a closed semicircular curve produced when first on the real axis of gradually integrated back from there in the semi- circular arc.

It is observed that disappears for the integral and thus

After the residue theorem is then

In order to avoid recurrent estimates for integrals of the form, you use the lemma of Jordan.

Examples

Example 1

It should be and. Here the Jordan lemma is applicable and it is

So is true for the integral of the real axis

Is split using the Euler's identity in the real and imaginary parts, we obtain the equality

Example 2

It should be. Analogously to Example 1, and thus

Proof of the lemma of Jordan

The integral can be written as the substitution. Estimate of the amount specified above gives

With. It follows

Since the integrand is with respect to axisymmetric. After Jordan 's inequality is for all, and therefore