Difference between revisions of "Jordan's lemma"

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(Created page with " We consider integrals of the form <math> \int_C f(z) e^{i a z} \; dz </math>")
 
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Here we follow standard texts, such as Spiegel (1964)<ref> Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).</ref> or Levinson and Redheffer (1970).  <ref> Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.</ref>
  
We consider integrals of the form  
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We consider integrals of the form <math> \int_C f(z) e^{i a z} \; dz </math>, where <math> C = C_1 + C_2 </math> is a closed simple contour as shown in Figure 1.
<math> \int_C f(z) e^{i a z} \; dz </math>
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The function <math> f(z) </math> has a finite number of poles <math> \{z_0,z_1,z_2,...,z_n\} </math> located inside the contour. As <math> |z| \rightarrow \infty</math> the function <math> |f(z)| </math> in the integrand is assumed to decay at least as fast as <math> \frac{1}{R}</math>, where <math> |z| = R</math> for points on <math>C_2</math>.
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[[File:contour_Jordan_3.png|thumb|center|350px|"Figure 1: the contour C for a Fourier-like integration (upward closure only)"]]
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By the [[Residue Theorem]] the value of this integral may be written as <math> 2 \pi i </math> times the sum of
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the residues of the integrand
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<center> <math> 2 \pi i \sum_{k=0}^n \mbox{Res}(f(z) e^{ i a z }; z= z_k )= \int_{C_1} f(z) e^{i a z} \; dz + \int_{C_2} f(z) e^{i a z} \; dz . </math> </center>
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Typically, the application of such a contour integral is to find the value of the integral along the real axis, which is the integral along <math> C_1 </math>.
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The next step is to evaluate the integral over the contour <math> C_2 </math> and show that this vanishes. We consider the following estimate
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<center> <math> \left|\int_{C_2} f(z) e^{i a z} \; dz \right| \le \int_{C_2}  \left| f(z) e^{i a z} \right| \;| dz | </math>. </center>
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By [[Cauchy's theorem]] we may assume that <math> C_2 </math> is a semi-circle, and may consider the polar form of <math> z = |z|e^{i\phi} </math>,
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<math> dz = i|z|e^{i \phi} d\phi </math>, and we may also consider <math> z = |z| (\cos \phi + i \sin \phi)</math> in the exponential. Substituting
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these into the  <math> C_2 </math> and applying the fact that as <math> R \rightarrow \infty</math> the function <math> |f(z)| \le \frac{M}{R} </math> yielding
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<center> <math> \int_{C_2}  \left| f(z) e^{i a z} \right| \;| dz | = \int_0^{\pi} \left|f(z) e^{i a |z|(\cos \phi  - \sin \phi)}\right| | i|z|e^{i \phi} d\phi| \le \lim_{R\rightarrow \infty} \frac{M}{R} \int_0^{\pi} e^{-aR \sin(\phi)} \; R d\phi,  </math> </center>
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where <math> R </math> is the radius of the semi-circle <math> C_2 .</math> For the integral to exist, the integrand must decay sufficiently rapidly. Thus,  the <math> \mbox{Re}\; a > 0. </math>Here, the [[Maximum Modulus theorem]] has been applied to generate the estimate of <math> |f(z)| \le M. </math>
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We may further proceed with the estimate by recognizing that integral is twice the same integral, with upper integration
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limit of <math> \frac{\pi}{2}</math> and the extra factor of 2 is absorbed into the <math> M </math> yielding
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<center> <math> \int_{C_2}  | f(z) e^{i a z}| \;| dz | \le \lim_{R\rightarrow \infty} M \int_0^{\pi/2} e^{-aR \sin(\phi)} \; d\phi \le \lim_{R\rightarrow \infty} M \int_0^{\pi/2} e^{-aR \frac{2}{\pi}(\phi)} \; d\phi </math> </center>
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<center> <math> \qquad \qquad \qquad \qquad \qquad = \lim_{R\rightarrow \infty}
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\left. \frac{ - 2 M}{\pi a R}  e^{-aR \frac{2}{\pi}(\phi)} \right|_0^{\pi/2} = \frac{2 M}{\pi a R} ( 1 - e^{-aR} ) \rightarrow 0 . </math> </center>
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Here we have used the fact that on the interval <math> [0, \pi/2 ]</math>  we have <math> -\sin(\phi) \le - \phi (\pi/2) </math>.
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The case of closure in the lower half plane of <math> z </math> follows the same way, with the exception that the
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<math> \mbox{Re}\; a < 0 </math>.
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Thus, the integral over <math> C_1 </math> is given by the expression
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<center> <math> 2 \pi i \sum_{k=0}^n \mbox{Res}( f(z) e^{i a z}; z=z_k) = \int_{C_1} f(z) e^{i a z} \; dz. </math> </center>
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=References =
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{{reflist}}
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== External links ==
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{{search}}

Latest revision as of 08:03, 12 October 2015

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

We consider integrals of the form , where is a closed simple contour as shown in Figure 1. The function has a finite number of poles located inside the contour. As the function in the integrand is assumed to decay at least as fast as , where for points on .

"Figure 1: the contour C for a Fourier-like integration (upward closure only)"

By the Residue Theorem the value of this integral may be written as times the sum of the residues of the integrand

Typically, the application of such a contour integral is to find the value of the integral along the real axis, which is the integral along .

The next step is to evaluate the integral over the contour and show that this vanishes. We consider the following estimate

.

By Cauchy's theorem we may assume that is a semi-circle, and may consider the polar form of , , and we may also consider in the exponential. Substituting these into the and applying the fact that as the function yielding

where is the radius of the semi-circle For the integral to exist, the integrand must decay sufficiently rapidly. Thus, the Here, the Maximum Modulus theorem has been applied to generate the estimate of

We may further proceed with the estimate by recognizing that integral is twice the same integral, with upper integration limit of and the extra factor of 2 is absorbed into the yielding

Here we have used the fact that on the interval we have .

The case of closure in the lower half plane of follows the same way, with the exception that the .

Thus, the integral over is given by the expression

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.

External links

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Jordan's lemma
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