|
|
| Line 1: |
Line 1: |
| In [[complex analysis]], '''Jordan's lemma''' is a result frequently used in conjunction with the [[residue theorem]] to evaluate [[contour integral]]s and [[improper integral]]s. It is named after the French mathematician [[Camille Jordan]].
| | The writer is known as Irwin. Bookkeeping is her day occupation now. For years he's been residing in North Dakota and his family loves it. Playing baseball is the hobby he will never quit doing.<br><br>Here is my site :: at home std testing - [http://livestudentjobs.co.uk/groups/solid-advice-for-handling-a-candida/ our homepage], |
| | |
| ==Statement==
| |
| Consider a [[complex number|complex]]-valued, [[continuous function]] ''f'', defined on a semicircular contour
| |
| | |
| :<math>C_R=\{z : z=R e^{i \theta}, \theta\in [0,\pi]\}</math>
| |
| | |
| of radius ''R'' > 0 lying in the upper half-plane, centred at the origin. If the function ''f'' is of the form
| |
| | |
| :<math>f(z)=e^{iaz} g(z)\,,\quad z\in C_R,</math>
| |
| | |
| with a parameter ''a'' > 0, then Jordan's lemma states the following upper bound for the contour integral:
| |
| | |
| :<math>\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|\,.</math>
| |
| | |
| where equal sign is when ''g(z)'' is identically zero. 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<sub>R</sub>'' for all large ''R'' and
| |
|
| |
| ::<math>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(*)</math>
| |
| | |
| :then by Jordan's lemma
| |
|
| |
| ::<math>\lim_{R \to \infty} \int_{C_R} f(z)\, dz = 0.</math>
| |
|
| |
| * 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<sub>R</sub>''.
| |
| | |
| ==Application of Jordan's lemma==
| |
| [[Image:Jordan's Lemma.svg|right|thumb|300px|The path ''C'' is the concatenation of the paths ''C''<sub>1</sub> and ''C''<sub>2</sub>.]]
| |
| Jordan's lemma yields a simple way to calculate the integral along the real axis of functions {{nowrap|''f'' (''z'') {{=}} ''e<sup>iaz</sup>g''(''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 ''z''<sub>1</sub>, ''z''<sub>2</sub>, ..., ''z<sub>n</sub>''. Consider the closed contour ''C'', which is the concatenation of the paths ''C''<sub>1</sub> and ''C''<sub>2</sub> shown in the picture. By definition,
| |
| | |
| :<math>\oint_{C} f(z)\, dz = \int_{C_1}f(z)\,dz + \int_{C_2} f(z)\,dz\,.</math>
| |
| | |
| Since on ''C''<sub>2</sub> the variable ''z'' is real, the second integral is real:
| |
| | |
| :<math>\int_{C_2} f(z)\,dz = \int_{-R}^{R} f(x)\,dx\,.</math> | |
| | |
| The left-hand side may be computed using the [[residue theorem]] to get, for all ''R'' larger than the maximum of |''z''<sub>1</sub>|, |''z''<sub>2</sub>|, ..., |''z<sub>n</sub>''|,
| |
| | |
| :<math>\oint_{C} f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{Res}(f, z_k)\,,</math>
| |
| | |
| where {{nowrap|Res(''f'', ''z<sub>k</sub>'')}} denotes the [[Residue (complex analysis)|residue]] of ''f'' at the singularity ''z<sub>k</sub>''. Hence, if ''f'' satisfies condition (*), then taking the limit as ''R''  tends to infinity, the contour integral over ''C''<sub>1</sub> vanishes by Jordan's lemma and we get the value of the improper integral
| |
| | |
| :<math>\int_{-\infty}^{\infty} f(x)\,dx = 2\pi i \sum_{k=1}^n \operatorname{Res}(f, z_k)\,.</math>
| |
| | |
| ==Example==
| |
| The function
| |
| | |
| :<math>f(z)=\frac{e^{iz}}{1+z^2},\qquad z\in{\mathbb C}\setminus\{i,-i\},</math>
| |
| | |
| satisfies the condition of Jordan's lemma with ''a'' = 1 for all ''R'' > 0 with ''R'' ≠ 1. Note that, for ''R'' > 1,
| |
| | |
| :<math>M_R=\max_{\theta\in[0,\pi]}\frac1{|1+R^2e^{2i\theta}|}=\frac1{R^2-1}\,,</math>
| |
| | |
| hence (*) holds. Since the only singularity of ''f'' in the upper half plane is at ''z'' = ''i'', the above application yields
| |
| | |
| :<math>\int_{-\infty}^\infty \frac{e^{ix}}{1+x^2}\,dx=2\pi i\,\operatorname{Res}(f,i)\,.</math>
| |
| | |
| Since ''z'' = ''i'' is a [[simple pole]] of ''f'' and {{nowrap|1 + ''z''<sup>2</sup>}} = {{nowrap|(''z'' + ''i'')(''z'' - ''i'')}}, we obtain
| |
| :<math>\operatorname{Res}(f,i)=\lim_{z\to i}(z-i)f(z)
| |
| =\lim_{z\to i}\frac{e^{iz}}{z+i}=\frac{e^{-1}}{2i}</math>
| |
| | |
| so that
| |
| | |
| :<math>\int_{-\infty}^\infty \frac{\cos x}{1+x^2}\,dx=\operatorname{Re}\int_{-\infty}^\infty \frac{e^{ix}}{1+x^2}\,dx=\frac{\pi}{e}\,.</math>
| |
| | |
| 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 [[Line_integral#Complex_line_integral|complex line integral]],
| |
| | |
| :<math>\begin{align}
| |
| \int_{C_R} f(z)\, dz
| |
| &=\int_0^\pi g(Re^{i\theta})\,e^{iaR(\cos\theta+i \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}</math>
| |
| | |
| Now the inequality
| |
| | |
| :<math>\biggl|\int_a^b f(x)\,dx\biggr|\le\int_a^b |f(x)|\,dx</math>
| |
| | |
| yields
| |
| | |
| :<math>\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}</math>
| |
| | |
| Using ''M<sub>R</sub>'' as defined in (*) and the symmetry {{nowrap|sin ''θ''}} = {{nowrap|sin(''π'' – ''θ'')}}, we obtain
| |
| | |
| :<math> 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\,.</math>
| |
| | |
| Since the graph of {{nowrap|sin ''θ''}} is [[concave function|concave]] on the interval {{nowrap|''θ'' ∈ <nowiki>[</nowiki>0,''π'' /2<nowiki>]</nowiki>}}, the graph of {{nowrap|sin ''θ''}} lies above the straight line connecting its endpoints, hence
| |
| | |
| :<math>\sin\theta\ge \frac{2\theta}{\pi}\quad</math>
| |
| | |
| for all {{nowrap|''θ'' ∈ <nowiki>[</nowiki>0,''π'' /2<nowiki>]</nowiki>}}, which further implies
| |
| | |
| :<math>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\,.</math>
| |
| | |
| ==See also==
| |
| *[[Estimation lemma]]
| |
| | |
| ==References==
| |
| * {{Cite book| last1=Brown| first1=James W.| last2=Churchill| first2=Ruel V.| date=2004| title=Complex Variables and Applications| edition=7th|place=New York | publisher=McGraw Hill| pages=262–265| isbn = 0-07-287252-7}}
| |
| | |
| {{DEFAULTSORT:Jordan's Lemma}}
| |
| [[Category:Complex analysis]]
| |
| [[Category:Articles containing proofs]]
| |
| [[Category:Lemmas]]
| |
The writer is known as Irwin. Bookkeeping is her day occupation now. For years he's been residing in North Dakota and his family loves it. Playing baseball is the hobby he will never quit doing.
Here is my site :: at home std testing - our homepage,