Contour Integration
المؤلف:
Arfken, G
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press
الجزء والصفحة:
...
17-11-2018
2146
Contour Integration
Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex residues inside the contour.
Let 
and 
be polynomials of polynomial degree 
and 
with coefficients 
, ..., 
and 
, ..., 
. Take the contour in the upper half-plane, replace 
by 
, and write 
. Then
 |
(1)
|
Define a path 
which is straight along the real axis from 
to 
and make a circular half-arc to connect the two ends in the upper half of the complex plane. The residue theorem then gives
where ![Res[z]](http://mathworld.wolfram.com/images/equations/ContourIntegration/Inline24.gif)
denotes the complex residues. Solving,
![lim_(R->infty)int_(-R)^R(P(z)dz)/(Q(z))=2piisum_(I[z]>0)Res(P(z))/(Q(z))-lim_(R->infty)int_0^pi(P(Re^(itheta)))/(Q(Re^(itheta)))iRe^(itheta)dtheta.](http://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation2.gif) |
(5)
|
Define
and set
 |
(10)
|
then equation (9) becomes
 |
(11)
|
Now,
 |
(12)
|
for 
. That means that for 
, or 
, 
, so
![int_(-infty)^infty(P(z)dz)/(Q(z))=2piisum_(I[z]>0)Res[(P(z))/(Q(z))]](http://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation6.gif) |
(13)
|
for 
. Apply Jordan's lemma with 
. We must have
 |
(14)
|
so we require 
.
Then
![int_(-infty)^infty(P(z))/(Q(z))e^(iaz)dz=2piisum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]](http://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation8.gif) |
(15)
|
for 
and 
. Since this must hold separately for real and imaginary parts, this result can be extended to
![int_(-infty)^infty(P(x))/(Q(x))cos(ax)dx=2piR<span style=]() {sum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]} " src="http://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation9.gif" style="height:47px; width:307px" /> |
(16)
|
![int_(-infty)^infty(P(x))/(Q(x))sin(ax)dx=2piI<span style=]() {sum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]}. " src="http://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation10.gif" style="height:47px; width:302px" /> |
(17)
|
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 406-409, 1985.
Krantz, S. G. "Applications to the Calculation of Definite Integrals and Sums." §4.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 51-63, 1999.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 353-356, 1953.
Whittaker, E. T. and Watson, G. N. "The Evaluation of Certain Types of Integrals Taken Between the Limits 
and 
," "Certain Infinite Integrals Involving Sines and Cosines," and "Jordan's Lemma." §6.22-6.222 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 113-117, 1990.
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