Let $f(z) = \textrm{Log}(\textrm{Log} (z + 2i))$. Where is $f$ continuous?
I have been having issues on determining the continuity of this double logarithmic complex function.
Do I approach this testing limits or derivatives of the function?
If anyone could help me here that would be great. Thanks
Let $w=\textrm{Log}(z+2i), \quad f(z) = \textrm{Log}(w(z))$.
We want (I.) $z+2i \notin \{-\infty,0]$ and (II.) $w \notin \{-\infty,0]$.
I. $$z \ne\alpha-2i, \quad \alpha \in (-\infty,0]$$
II. $$\begin{aligned} \textrm{Log}(z+2i) &\notin (-\infty,0] \\ z+2i &\ne e^\alpha, \quad \alpha \in (-\infty,0]\\ z &\ne \beta -2i, \quad \beta \in [0,1] \end{aligned} $$
In summary, for $f(z)$ to be defined, we want $z \notin \textrm{I} \cup \textrm{II}$:
$$z\ne \alpha -2i, \quad \alpha \in (-\infty, 1]$$
Since the logarithm is holomorphic where it is defined, this is also the condition for $f(z)$ to be continuous.