6.3 • THE CAUCHY-GOURSAT THEOREM 221- EXAMPLE 6.12 Recall that expz, cosz, and zn (where n is a positive inte-
ger) are all entire functions. The Cauchy-Goursat theorem implies that, for any
simple closed contour,
lo expz dz= 0, lo cosz dz= 0, and lo z" dz= 0.
- EXAMPLE 6.13 Let n be an integer. If C is a simple closed contour such
that the origin does not lie interior to C, then there is a simply connected domain
D that contains C in which f (z) = ,~ is analytic, as is indicated in Figure 6.22.
The Cauchy-Goursat theorem implies that fc ,^1 .. dz= 0.
We want to be able to replace integrals over certain complicated contours
with integrals that are easy to evaluate. If C 1 is a simple closed contour that
can be "continuously deformed" into another simple closed contour C 2 without
passing through a point where f is not analytic, then the value of the contour
integral off over C 1 is the same as the value of the integral off over C2. To be
precise, we state the following result.