# Show that $\log\log z$ is analytic

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!

• 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? Oct 28, 2014 at 15:21
• What is the image $\operatorname{Log} (\mathbb{C}\setminus (-\infty,1])$? Oct 28, 2014 at 15:44

## 2 Answers

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

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 $$g(z)$$.