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. The theorem is named after the French mathematician Camille Jordan.

1 Statement
- 1.1 Remarks
2 Application of Jordan's lemma
3 Example
4 Proof of Jordan's lemma
5 See also
6 References

Statement

Consider a complex-valued, continuous function f, defined on a semicircular contour

$C_R=\{z�: z=R e^{i \theta}, \theta\in [0,\pi]\}$

of radius R > 0 lying in the upper half-plane, centred at the origin. If the function f is of the form

$f(z)=e^{iaz} g(z)\,,\quad z\in C_R,$

with a parameter a > 0, then Jordan's lemma states the following upper bound for the contour integral:

$\biggl|\int_{C_R} f(z)\, dz\biggr| \le \frac\pi{a}\max_{\theta\in [0,\pi]} \bigl|g \bigl(R e^{i \theta}\bigr)\bigr|\,.$

An analogous statement for a semicircular contour in the lower half-plane holds when a < 0.

Remarks

If f is defined and continuous on the semicircular contour C_R for all large R and

$M_R:=\max_{\theta\in [0,\pi]} \bigl|g \bigl(R e^{i \theta}\bigr)\bigr| \to 0\quad \mbox{as } R \to \infty\,,\qquad(*)$

then by Jordan's lemma

$\lim_{R \to \infty} \int_{C_R} f(z)\, dz = 0.$

For the case a = 0, see the estimation lemma.

Compared to the estimation lemma, the upper bound in Jordan's lemma does not explicitly depend on the length of the contour C_R.

Application of Jordan's lemma

Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f (z) = e^iazg(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, ..., z_n. Consider the closed contour C, which is the concatenation of the paths C₁ and C₂ shown in the picture. By definition,

$\oint_{C} f(z)\, dz = \int_{C_1}f(z)\,dz %2B \int_{C_2} f(z)\,dz\,.$

Since on C₂ the variable z is real, the second integral is real:

$\int_{C_2} f(z)\,dz = \int_{-R}^{R} f(x)\,dx\,.$

The left-hand side may be computed using the residue theorem to get, for all R larger than the maximum of |z₁|, |z₂|, ..., |z_n|,

$\oint_{C} f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{Res}(f, z_k)\,,$

where Res(f, z_k) denotes the residue of f at the singularity z_k. 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

$\int_{-\infty}^{\infty} f(x)\,dx = 2\pi i \sum_{k=1}^n \operatorname{Res}(f, z_k)\,.$

Example

The function

$f(z)=\frac{e^{iz}}{1%2Bz^2},\qquad z\in{\mathbb C}\setminus\{i,-i\},$

satisfies the condition of Jordan's lemma with a = 1 for all R > 0 with R ≠ 1. Note that, for R > 1,

$M_R=\max_{\theta\in[0,\pi]}\frac1{|1%2BR^2e^{2i\theta}|}=\frac1{R^2-1}\,,$

hence (*) holds. Since the only singularity of f in the upper half plane is at z = i, the above application yields

$\int_{-\infty}^\infty \frac{e^{ix}}{1%2Bx^2}\,dx=2\pi i\,\operatorname{Res}(f,i)\,.$

Since z = i is a simple pole of f and 1 + z² = (z + i)(z - i), we obtain

$\operatorname{Res}(f,i)=\lim_{z\to i}(z-i)f(z) =\lim_{z\to i}\frac{e^{iz}}{z%2Bi}=\frac{e^{-1}}{2i}$

so that

$\int_{-\infty}^\infty \frac{\cos x}{1%2Bx^2}\,dx=\operatorname{Re}\int_{-\infty}^\infty \frac{e^{ix}}{1%2Bx^2}\,dx=\frac{\pi}{e}\,.$

This result exemplifies how some integrals difficult to compute with classical tools are easily tackled with the help of complex analysis.

Proof of Jordan's lemma

By definition of the complex line integral,

$\begin{align} \int_{C_R} f(z)\, dz &=\int_0^\pi g(Re^{i\theta})\,e^{iaR(\cos\theta%2Bi \sin\theta)}\,i Re^{i\theta}\,d\theta\\ &=R\int_0^\pi g(Re^{i\theta})\,e^{aR(i\cos\theta-\sin\theta)}\,ie^{i\theta}\,d\theta\,. \end{align}$

Now the inequality

$\biggl|\int_a^b f(x)\,dx\biggr|\le\int_a^b |f(x)|\,dx$

yields

$\begin{align} I_R:=\biggl|\int_{C_R} f(z)\, dz\biggr| &\le R\int_0^\pi\bigl|g(Re^{i\theta})\,e^{aR(i\cos\theta-\sin\theta)}\,ie^{i\theta} \bigr|\,d\theta\\ &=R\int_0^\pi \bigl|g(Re^{i\theta})\bigr|\,e^{-aR\sin\theta}\,d\theta\,. \end{align}$

Using M_R as defined in (*) and the symmetry sin θ = sin(π – θ), we obtain

$I_R \le RM_R\int_0^\pi e^{-aR\sin\theta}\,d\theta = 2RM_R\int_0^{\pi/2} e^{-aR\sin\theta}\,d\theta\,.$

Since the graph of sin θ is concave on the interval θ ∈ [0,π /2], the graph of sin θ lies above the straight line connecting its endpoints, hence

$\sin\theta\ge \frac{2\theta}{\pi}\quad$

for all θ ∈ [0,π /2], which further implies

$I_R \le 2RM_R \int_0^{\pi/2} e^{-2aR\theta/\pi}\,d\theta =\frac{\pi}{a} (1-e^{-a R}) M_R\le\frac\pi{a}M_R\,.$

References

Brown, James W.; Churchill, Ruel V. (2004). Complex Variables and Applications (7th ed.). New York: McGraw Hill. pp. 262–265. ISBN 0-07-287252-7.