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Show that $\log\log z$ is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0$, $x \le 1$.

As of now I'm not too sure on how to solve this problem, so I was thinking you may have to use the Cauchy-Riemann equations to find the answer. I honestly tried it but I don't know what to do.

If someone can help me out in solving this problem that would be great. Thanks!

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    $\begingroup$ Can you make use of tha fact that the derivative $\frac1z\cdot\frac1{\log z}$ is analytic in that region and the region is simply connected? $\endgroup$ Oct 28, 2014 at 15:21
  • $\begingroup$ What is the image $\operatorname{Log} (\mathbb{C}\setminus (-\infty,1])$? $\endgroup$ Oct 28, 2014 at 15:44

2 Answers 2

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I think you have you to use the principal value of the logarithm. $\log z = \log |z| + \mathrm i\arg z$.

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Let $$\begin{aligned} f(z)&=\log z, \quad z \in \mathbb{C}\backslash(-\infty, 0], \\ g(z)&=\log z, \quad z \in \mathbb{C}\backslash(-\infty, 1] = \mathbb{C}\backslash \left\{ (-\infty,0]\cup (0,1]\right\} \end{aligned} $$

Both $f(z)$ and $g(z)$ are holomorphic in their domains.

The image of $(0,1]$ under $f(z)$ is $(-\infty,0]$.

By removing $(0,1]$ from the domain of $w=g(z)$, then $f(g(z)) = f(w)=\log w = \log \log z$ is holomorphic over the domain of $g(z)$.

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