is there anybody there who is interested in complex calculus?
Is the theorem right?
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When you integrate f on a closed loop, you may or may not get 0 as the total integral. It depends a lot on what f is, and what the loop is. If f has an anti-derivative on the loop, then the integral is guaranteed to be 0 by the fundamental theorem of calculus. If f is analytic on the loop and inside the loop, then Cauchy's theorem guarantees that the integral is 0. In all other cases it is very unlikely that the integral vanishes completely.
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z=2exp(I@)
f(z)=Logz=lnr+i@ then the antiderivative is exist is 1/z and by calculation the integral yields "2pi*i". But according to the theorem above shouldn't it be 0.
Where am I wrong?

I remember this vaguely from my university. I believe Log(z) is not defined on the whole set C. I believe there is a line going from (0, 0) under the angle "fi" (for different Log(z) functions or something) that Log(z) is not defined for points on that line. Am I correct? If true, and if your loop intersects that line, I believe that might be causing the problem. Then again, I remember that an integral is not affected if a function is not defined in a single point, but I don't know if that means anything. Is that any help?