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Thread: complex analytic functions

  1. #1
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    complex analytic functions

    Prove 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.
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  2. #2
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    If $\displaystyle f(z)$ is analytic on an (non-trivial) open set $\displaystyle S$ and $\displaystyle e^{f(z)} = z$ for all point $\displaystyle z\in S$ then $\displaystyle f(z) = \ln |z| + i\arg(z)$ for some particular definition of $\displaystyle \arg z$. Thus, our goal is to prove that $\displaystyle F(z)$ as you defined it above has the property that $\displaystyle e^{F(z)} = z$ but then that would mean $\displaystyle F(z) = \ln |z| + i \arg(z)$ for some particular definition of of the argument, but then this contradicts the fact that $\displaystyle F$ is analytic on $\displaystyle 1<|z|<2$ because the branch of $\displaystyle \arg z$ will have to hit the annulus as it leaves the origin. There is just one detail I do not know how to show, maybe it is not needed: there is a point $\displaystyle z_0 $ on the annulus so that $\displaystyle e^{F(z_0)} = z_0$. With that we can complete the proof. Suppose that $\displaystyle F'(z) = 1/z$ and consider $\displaystyle g(z) = ze^{-F(z)}$ then $\displaystyle g'(z) = e^{-F(z)} - z(1/z)e^{-F(z)} = 0$ and this means $\displaystyle g(z) = k$ for some complex number $\displaystyle k$ because the annulus is a region (open connected set). This means, $\displaystyle ze^{-F(z)} = k$ for all point on the annulus. In particular for $\displaystyle z_0$, which yields. $\displaystyle z_0e^{-F(z_0)} = k \implies k=1$. Thus, $\displaystyle ze^{-F(z)} = 1 \implies e^{F(z)} = z$. We have show $\displaystyle F(z)$ is the inverse function of $\displaystyle e^z$ and this completes the proof using the begining comments.
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