Advanced Calculus fi..

Rule IV could also be used, with A = eZ, B = z3 - 2z2 + z:Chapter 8 Functions of a Complex Variable 579Accordingly,LU -- - "8.16 RESIDUE AT INFINITYLet f (z) be analytic for Izl > R. The residue offz) at oo is defined as follows:where the integral is taken in the negative direction on a simple closed path C, inthe domain of analyticity off (z), and outside of which f (z) has no singularity otherthan oo. This is suggested in Fig. 8.18. Theorem 27 has an immediate extension tothis case:THEOREM 29whereThe residue of f (z) at oo is given by the equationRes [ f (z),ool = -a-I, (8.78)is the coefficient of z-' in the Laurent expansion of f (z) at CQ:f(z)= ...+--+...a-n+-+ao+alz+.-..a-1(8.79)zn zThe proof is the same as for Theorem 27. It should be stressed that the presenceof a nonzero residue at oo is not related to presence of a pole or essential singularityat oo. That is, f (2) can have a nonzero residue whether or not there is a pole oressential singularity, for the pole or essential singularity at oo is due to the positivepowers of z, not to negative powers (Section 8.14). Thus the function ellz = 1 +z-' + (2!z2)-I + . is analytic at CQ but has the residue - 1 there.Figure 8.18Residue at infinity.