$f(z) = \textrm{Log}(\textrm{Log }(z + 2i))$. Where is $f$ defined?

Question

I am having trouble with this complex function.

Let $f(z) = \textrm{Log}(\textrm{Log} (z + 2i))$. Where is $f$ defined?

If $\textrm{Log}(z)=\log|z|+i\textrm{Arg}(z)$.

If anyone could show me where this function is defined that would be great as I will be able to investigate the continuity of it as a result. Thanks.

Answer 1

The principal branch of the logarithm has a cut along the negative real axis. Thus $z+2i \ne -a$ where $a$ is a non-negative real number. $z \ne -a-2i$. If $w=\textrm{Log} (z+2i)$, then $w \ne -b$ where $b$ is a non-negative real.

$\textrm{Log} (z+2i) = \ln x^2 + (y+2)^2 + i \textrm{Arg} (z+2i) \Rightarrow x^2+(y+2)^2$ is not less than or equal to 1 with $\arctan \frac{y+2}{x} = 0$.

This can occur if $y=-2$ and $x\le1$.

In summary: $$(x,y) \notin \{ (\alpha, -2):\alpha \le 1 \} $$

Answer 2

HINT

think about what can make your function undefined. For example, if $z$ is in the unit circle, you get $|z| < 1$ which mean $\log |z| < 0$. What would happen to $f(z)$?

Are there other conditions that may make it undefined?