Page 53 - Calculus with Complex Numbers

Ffgure 4.8

Theorem 5 (Residue theorem) lf y is a closed contour and if flz) is diff-
erentiable on y and irlside y except at cl , . . . , cn inside y , then

where Rk is the residue of flz) at ck.

Proof (Special case) Suppose f (z) has a single singularity at z = c irlside y . lf
we let yr be a circle centre c radius r small enough to ensure that yr lies inside y
then by Corollal'y 2 of Cauchy's theorem we have

lf the Laurent expansion of flz) at z = c is

then we have

since for n # - 1 we have
(Z - C)X-F1
U - c)n dz =
r n + 1
r p'r
by M ethod 2.