Let $\Re(z) $ and $\Im(z)$ denote the real and imaginary part of $z \in \mathbb{C}$ respectively. Consider the domain $D=\{z\in \mathbb{C}: \Re(z)>|\Im(z)|\}$ and let $f_{n}(z) =\log(z^{n})$ where $n\in \{1,2,3,4\} $ and where $\log$ is the principle branch logarithm defined from $\mathbb{C}-(-\infty, 0]$ to $\mathbb{C}.$ Then which of the following are true?
A) $f_{1}(D) = \{z\in \mathbb{C}: 0\leqslant |\Im(z)| < \dfrac{\pi}{4}\}$
B) $f_{2}(D) = \{z\in \mathbb{C}: 0\leqslant |\Im(z)| < \dfrac{\pi}{2}\}$
C) $f_{3}(D) = \{z\in \mathbb{C}: 0\leqslant |\Im(z)|< \dfrac{3\pi}{4}\}$
B) $f_{4}(D) = \{z\in \mathbb{C}: 0\leqslant |\Im(z)| < \pi\}$
I try to conclude something from given domain D like $\Re(z)>|\Im(z)|$ is given so $|\Im(z)|<\dfrac{r}{\sqrt{2}}$ and try to do something from $\log z=\ln r+i\operatorname{Arg}z $. I did it only for $f_{1}$ to check first is true or not but from this I can't decide it.
So Please help to solve this.
They are all true.
Since the imaginary part of the logarithm function is the argument $$ \log z^n = \log |z|^n + i n \arg(z),$$ and since the allowable values for $z^n$ are $$-\pi < \arg z^n < \pi$$
it follows that (for $n \in \{ 1,2,3,4\}$)
$$f_n(D) = \left \{z\in \mathbb{C}: -\frac{n \pi}{4} <\Im(z^n) < \dfrac{n \pi}{4}\right \}$$
For each $n \in \{1,2,3,4\}$, the corresponding statement is equivalent.