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 P(x) and Q(x) be polynomials of polynomial degree n and m with coefficients b_n, ..., b_0 and c_m, ..., c_0. Take the contour in the upper half-plane, replace x by z, and write z=Re^(itheta). Then
Define a path gamma_R which is straight along the real axis from -R to R and make a circular half-arc to connect the two ends in the upper half of the complex plane. The residue theorem then gives
![画像:2piisum_(I[z]>0)Res[(P(z))/(Q(z))],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ContourIntegration/Inline23.svg){kind=link}
where Res[z] 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. |
(5)
|
Define
and set
| epsilon=-(n+1-m), |
(10)
|
then equation (9) becomes
Now,
| lim_(R->infty)R^(-epsilon)=0 |
(12)
|
for epsilon>0. That means that for -n-1+m>=1, or m>=n+2, I_R=0, so
![画像: int_(-infty)^infty(P(z)dz)/(Q(z))=2piisum_(I[z]>0)Res[(P(z))/(Q(z))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation6.svg){kind=link}
for m>=n+2. Apply Jordan's lemma with f(x)=P(x)/Q(x). We must have
| lim_(x->infty)f(x)=0, |
(14)
|
so we require m>=n+1.
Then
![画像: int_(-infty)^infty(P(z))/(Q(z))e^(iaz)dz=2piisum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation8.svg){kind=link}
for m>=n+1 and a>0. Since this must hold separately for real and imaginary parts, this result can be extended to
![画像: int_(-infty)^infty(P(x))/(Q(x))cos(ax)dx=2piR{sum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation9.svg){kind=link}
![画像: int_(-infty)^infty(P(x))/(Q(x))sin(ax)dx=2piI{sum_(I[z]>0)Res[(P(z))/(Q(z))e^(iaz)]}.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ContourIntegration/NumberedEquation10.svg){kind=link}
See also
Cauchy Integral Formula, Cauchy Integral Theorem, Contour, Contour Integral, Complex Residue, Inside-Outside Theorem, Jordan's Lemma, Sine IntegralExplore with Wolfram|Alpha
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 -infty and +infty," "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.Referenced on Wolfram|Alpha
Contour IntegrationCite this as:
Weisstein, Eric W. "Contour Integration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ContourIntegration.html