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About LinkBacksProve there exists no function F(z) analytic in the annulus D: 1<|z|<2 such that F'(z) = 1/z.
I assumed there was such a function.
let F(z)=Log(z) + c
F'(z)= 1/z
I don't know how to do this.
Ifis analytic on an (non-trivial) open set
and
for all point
then
for some particular definition of
. Thus, our goal is to prove that
as you defined it above has the property that
but then that would mean
for some particular definition of of the argument, but then this contradicts the fact that
is analytic on
because the branch of
on the annulus so that
. With that we can complete the proof. Suppose that
and consider
then
and this means
for some complex number
because the annulus is a region (open connected set). This means,
for all point on the annulus. In particular for
, which yields.
. Thus,
. We have show
and this completes the proof using the begining comments.
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