# 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 $y = 0, x ≤ 1$.

As of now im 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? – Hagen von Eitzen Oct 28 '14 at 15:21
• What is the image $\operatorname{Log} (\mathbb{C}\setminus (-\infty,1])$? – Daniel Fischer Oct 28 '14 at 15:44

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