Re: Residue Function

*To*: mathgroup at smc.vnet.net*Subject*: [mg76303] Re: Residue Function*From*: CKWong <CKWong.P at gmail.com>*Date*: Sat, 19 May 2007 04:41:30 -0400 (EDT)*References*: <f2bulj$jhe$1@smc.vnet.net><f2h9g6$185$1@smc.vnet.net>

It seems appropriate to describe briefly how the residue at infinity is calculated. Firstly, we can deal only with the case where a counterclockwise contour C exists that can encompass all the poles of the function in the finite complex plane. Next, we treat the complex plane as a sphere with 0 at the south pole and infinity at the north pole. ( This is just the reverse of the spherical projection of a sphere onto a plane. ) Now the contour C , when traveled clockwise, can be taken as enclosing the point infinity. Therefore, the residue at infinity is equal to the negative of the sum of the residues of all the poles in the finite complex plane. In the special case where there is only one pole in the finite plane, the residue at infinity is equal to the negative of that of the former. If C doesn't exist, there is no general way to calculate the residue at infinity.