Originally Posted by
chocaholic Explain in detail why the fact that the integral S(1/z)dz=1/(2(pi)i) taken on the unit circle with center zero i.e lzl=1 ensures that f(z)=1/z does not have an antiderivative on the region {z: z does not equal 0}.
my thoughts
f(z) is continuous on the region as the region does not include z=0 and the contour lzl=1 lies entirely in the region so the integral should be path independent and so the integral should have an antiderivative (i.e logz) and the integral around closed surves lying entirely in the region should all have values of 0 but 1/(2(pi)i) does not equal 0 so the antiderivative does not exist (also at (1,0) on the circle 1/(2(pi)i) is undefined).
Is this reasoning correct and complete? If not could someone please help me.
Thanks in advance.