Residue Theorem

An analytic function 👁 f(z)
whose Laurent series is given by

can be integrated term by term using a closed contour 👁 gamma
encircling 👁 z_0
,

The Cauchy integral theorem requires that the first and last terms vanish, so we have

where 👁 a_(-1)
is the complex residue. Using the contour 👁 z=gamma(t)=e^(it)+z_0
gives

so we have

If the contour 👁 gamma
encloses multiple poles, then the theorem gives the general result

where 👁 A
is the set of poles contained inside the contour. This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points inside the contour.

The diagram above shows an example of the residue theorem applied to the illustrated contour 👁 gamma
and the function

Only the poles at 1 and 👁 i
are contained in the contour, which have residues of 0 and 2, respectively. The values of the contour integral is therefore given by

See also

Cauchy Integral Formula, Cauchy Integral Theorem, Complex Residue, Contour, Contour Integral, Contour Integration, Group Residue Theorem, Laurent Series, Pole

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References

Knopp, K. "The Residue Theorem." §33 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 129-134, 1996.Krantz, S. G. "The Residue Theorem." §4.4.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 48-49, 1999.

Referenced on Wolfram|Alpha

Residue Theorem

Cite this as:

Weisstein, Eric W. "Residue Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ResidueTheorem.html

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