Show that $\log\log z$ is analytic

Question

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!

Answer 1

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

Answer 2

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 $g(z)$.