Lemma of Jordan

The Jordan lemma (after Marie Ennemond Camille Jordan ) is an aid to function theory . It is used in conjunction with the residual theorem to compute integrals from real analysis.

statement

If and converges uniformly to zero for all in the upper half-plane , then applies ${\displaystyle \alpha >0}$ ${\displaystyle g}$ ${\displaystyle |z|\to \infty }$

${\displaystyle \int _{K_{R}}g(z)\,e^{i\alpha z}dz\to 0}$

for . ${\displaystyle R\to \infty }$

This also applies if is and also tends to zero evenly in the upper half-plane. The lemma for the lower half-plane can be formulated completely analogously. ${\displaystyle \alpha =0}$ ${\displaystyle z\cdot g(z)}$

application

Integration path as a semicircular curve , which is closed by the real interval [-R, R] ${\displaystyle \gamma _{R}}$ ${\displaystyle K_{R}}$

Many improper integrals of the form , if they exist, can be calculated in the following way: One integrates on a closed semicircular curve , which arises when integrating first on the real axis from to and from there in the semicircular arc back to . ${\displaystyle \textstyle \int _{-\infty }^{\infty }f(z)\,dz}$ ${\displaystyle f}$ ${\displaystyle \gamma _{R}}$ ${\displaystyle -R}$ ${\displaystyle R}$ ${\displaystyle K_{R}}$ ${\displaystyle -R}$

It is found that for the integral vanishes and thus ${\displaystyle R\to \infty }$ ${\displaystyle \textstyle \int _{K_{R}}f\,dz}$

${\displaystyle \oint _{\gamma _{R}}fdz=\int _{[-R,R]}f\,dz+\int _{K_{R}}f\,dz{\xrightarrow[{R\to \infty }]{\ }}\int _{\mathbb {R} }f\,dz}$ applies.

After the residual theorem is then

${\displaystyle \int _{\mathbb {R} }f\,dz=\lim _{R\to \infty }\oint _{\gamma _{R}}fdz=2\pi i\sum _{\mathrm {Im} z>0}\mathrm {Res} f|_{z}}$.

In order to avoid recurring estimates for integrals of the form, one uses the Jordan lemma. ${\displaystyle \textstyle \int _{K_{R}}g(z)\,e^{i\alpha z}dz}$

Examples

1st example

It be and . The Jordan Lemma is applicable here and it holds ${\displaystyle g(z)={\tfrac {1}{1+z^{2}}}}$ ${\displaystyle f(z)=g(z)\,e^{i\alpha z}}$

${\displaystyle \lim _{R\to \infty }\int _{K_{R}}f(z)\,dz=0.}$

So it holds for the integral over the real axis

${\displaystyle \int _{\mathbb {R} }f(z)\,dz=2\pi i\,\mathrm {Res} f|_{i}=\pi \,e^{-\alpha }}$.

If you split up into real and imaginary parts with the help of Euler's identity , you get equality ${\displaystyle e^{i\alpha z}}$

${\displaystyle \int _{-\infty }^{\infty }{\frac {\cos(\alpha x)}{1+x^{2}}}\,dx=\pi \,e^{-\alpha }}$.

2nd example

Be it . Analogous to the 1st example is and thus ${\displaystyle g(z)={\tfrac {z}{1+z^{2}}}}$ ${\displaystyle \textstyle \int _{\mathbb {R} }f(z)\,dz=2\pi i\,\mathrm {Res} f|_{i}=i\pi \,e^{-\alpha }}$

${\displaystyle \int _{-\infty }^{\infty }{\frac {x\sin(\alpha x)}{1+x^{2}}}\,dx=\pi \,e^{-\alpha }}$.

Proof of the Jordan Lemma

The integral can be written after substitution as . Estimation of the amount upwards results ${\displaystyle \textstyle I_{R}:=\int _{K_{R}}g(z)\,e^{i\alpha z}\,dz}$ ${\displaystyle z=R\,e^{i\varphi }}$ ${\displaystyle \textstyle \int _{0}^{\pi }g\left(Re^{i\varphi }\right)\,e^{i\alpha Re^{i\varphi }}\,R\,e^{i\varphi }\,i\,d\varphi }$

${\displaystyle |I_{R}|\leq R\,\varepsilon _{R}\int _{0}^{\pi }e^{-\alpha R\sin \varphi }\,d\varphi }$

with . It follows ${\displaystyle \textstyle \varepsilon _{R}:=\max _{z\in K_{R}}|g(z)|}$

${\displaystyle |I_{R}|\leq 2R\,\varepsilon _{R}\int _{0}^{\frac {\pi }{2}}e^{-\alpha R\sin \varphi }\,d\varphi }$,

since the integrand is axially symmetric with respect to . According to Jordan’s inequality, is for all and therefore ${\displaystyle e^{-\alpha R\sin \varphi }}$ ${\displaystyle \varphi ={\tfrac {\pi }{2}}}$ ${\displaystyle \sin(\varphi )\geq {\tfrac {2}{\pi }}\,\varphi }$ ${\displaystyle \varphi \in \left[0,{\tfrac {\pi }{2}}\right]}$

${\displaystyle |I_{R}|\leq 2R\,\varepsilon _{R}\int _{0}^{\frac {\pi }{2}}e^{-\alpha R{\frac {2}{\pi }}\varphi }\,d\varphi ={\frac {\pi \,\varepsilon _{R}}{\alpha }}\left(1-e^{-\alpha R}\right)\leq {\frac {\pi \,\varepsilon _{R}}{\alpha }}\to 0}$for . ${\displaystyle R\to \infty }$

literature

• ED Solomentsev: Jordan Lemma . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).