# Jordan's Lemma

Here we follow standard texts, such as Spiegel (1964)[1] or Levinson and Redheffer (1970). [2]

In the computation of Fourier transform-like integrals as contour integrals, we often encounter the issue of the contribution of a semicircular contour that has infinite radius.

## Statement of Jordan's lemma

"Figure 1: contour of integration for Jordan's lemma"

Given a complex valued function ${\displaystyle f(z)}$ such that ${\displaystyle |f(z)|=o(1)}$ as ${\displaystyle |z|\rightarrow \infty }$. This means that ${\displaystyle |f(z)|\rightarrow 0}$ as ${\displaystyle |z|}$ becomes large. We require also that ${\displaystyle f(z)e^{iaz}}$ be analytic at infinity in some half-plane of ${\displaystyle z}$ we have the result that the integral over a semicircular contour in that half-plane vanishes

${\displaystyle \int _{C_{1}}f(z)e^{iaz}\;dz=0}$.

The quantity ${\displaystyle a}$ is real valued, and finite.

### Proof of Jordan's lemma

"Figure 2: the estimate of ${\displaystyle -\sin(\phi )}$ for Jordan's lemma"

Following the hypothesis of the lemma we consider the following contour integral

${\displaystyle \int _{C_{1}}f(z)e^{iaz}\;dz}$.

If we let ${\displaystyle R=|z|}$ and note by hypothesis that ${\displaystyle |f(z)|=o(1)}$ on ${\displaystyle C_{1}}$, we have

${\displaystyle \left|\int _{C_{1}}f(z)e^{iaz}\;dz\right|=\left|\int _{C_{1}}f(z)e^{iaR(\cos(\phi )+i\sin(\phi )}\;dz\right|\leq \int _{C_{1}}\left|f(z)e^{iaR(\cos(\phi )+i\sin(\phi ))}\right|\;|dz|}$

${\displaystyle \leq o(1)R\int _{0}^{\pi }e^{-aR\sin(\phi )}\;d\phi =o(1)\;2R\int _{0}^{\pi /2}e^{-aR\sin(\phi )}\;d\phi }$,

where the last equality follows because the integrand is symmetric about the point ${\displaystyle \phi =\pi /2}$.

The function ${\displaystyle -\sin(\phi )\leq -2/\pi \phi }$ on the interval ${\displaystyle [0,\pi /2]}$ allowing the further estimate (see Figure 2)

${\displaystyle o(1)2R\int _{0}^{\pi /2}e^{-aR\sin(\phi )}\;d\phi \leq o(1)2R\int _{0}^{\pi /2}e^{-aR2/\pi \phi }\;d\phi }$ .

The last integral may be solved by elementary means to yield

${\displaystyle o(1)2R\int _{0}^{\pi /2}e^{-aR2/\pi \phi }\;d\phi =\left.{\frac {o(1)}{a\pi }}e^{-aR2/\pi \phi }\right|_{0}^{\pi /2}=o(1)(1-e^{-aR})\rightarrow 0}$

as ${\displaystyle R\rightarrow \infty }$.

Though we have depicted the contour of integration in the upper half-plane of ${\displaystyle z}$, the result is actually general. The value of the parameter ${\displaystyle a}$ takes on the appropriate sign as to make the exponential always decaying. So, a similar result exists for closure of the contour in the lower half-plane of ${\displaystyle z}$, for Fourier-like integrals. Furthermore, for Laplace-like integrals we may consider closure in the right or left half-planes of the integration variable and obtain a similar result.

## References

1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.