Complex Analysis - Maths KU

APP-8 Appendix II Proof of the Cauchy-Goursat Theorem
The proof of the final part of the Cauchy-Goursat theorem demonstrates
that any closed contour C can be approximated to any desired degree of
accuracy by a closed polygonal path.
Theorem A.4 Any Simple Closed Contour
�If
C is a simple closed contour lying entirely within D, then
f(z) dz =0.
C
In Figure AII.5 we have shown a simple closed contour C and n points
z1, z2, ... , zn on C through which a polygonal curve P has been constructed.
Then it can be shown that the difference between the integral along
C, �
�
f(z) dz, and the integral along the polygonal contour P , f(z) dz,
C P
can � be made arbitrarily small as n →∞. As a consequence of Theorem A.3,
f(z) dz = 0 for any n, and thus the integral along C must also be zero.
P
C
z n–2
z n–1
P
z 5
z n
z 4
z 1
Figure AII.5 Simple closed contour C approximated by a closed polygonal curve P
z 3
z 2