Given a complex function , consider the Laurent Series
|
(1) |
Integrate term by term using a closed contour encircling ,
The Cauchy Integral Theorem requires that the first and last terms vanish, so we have
|
(3) |
But we can evaluate this function (which has a Pole at ) using the Cauchy Integral Formula,
|
(4) |
This equation must also hold for the constant function , in which case it is also true that , so
|
(5) |
|
(6) |
and (3) becomes
|
(7) |
The quantity is known as the Residue of at . Generalizing to a
curve passing through multiple poles, (7) becomes
|
(8) |
where is the Winding Number and the superscript denotes the quantity
corresponding to Pole .
If the path does not completely encircle the Residue, take the Cauchy
Principal Value to obtain
|
(9) |
If has only Isolated Singularities, then
|
(10) |
The residues may be found without explicitly expanding into a Laurent Series as follows:
|
(11) |
If has a Pole of order at , then for and
. Therefore,
|
(12) |
|
(13) |
Iterating,
|
|
|
(16) |
So
|
(17) |
and the Residue is
|
(18) |
This amazing theorem says that the value of a Contour Integral in the Complex Plane depends only on
the properties of a few special points inside the contour.
See also Cauchy Integral Formula, Cauchy Integral Theorem, Contour Integral, Laurent Series,
Pole, Residue (Complex Analysis)
© 1996-9 Eric W. Weisstein
1999-05-25