is there anyone there who are interested in complex integral?
is the theorem right?(should be)
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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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here is the question:
z=2exp(i@) (@=teta)
the principal branch--->Logz=lnr+i@
and f(z) has an antiderivative in the closed cloop wich is 1/z.
after the calculations the integral yields 2pi*i.BUt according the the theorem above it has should be 0. where am i wrong?