I was going through my textbook in Complex Analysis and I found the following exercise:
Let $D\subset\mathbb{C}$ be a domain and $f=u+iv$ be a holomorphic function on $D$. Which of the following functions are harmonic?
(a) $a=\dfrac{u}{\sqrt{u^2+v^2}}$
(b) $b=\dfrac{u}{u^2+v^2}$
(c) $c=\dfrac{u-v}{u^2+v^2}$
I know that a function $h$ is harmonic if and only if there is a holomorphic function $g$ such that $\text{Re} \ g = h$ (or at least this holds locally). This means that $u$ is harmonic, i.e. $\Delta u = 0$. Firstly, I tried to calculate the Laplacian of these functions ($\Delta a, \Delta b, \Delta c$) but these calculations turn out pretty ugly. So instead I tried to find holomorphic functions $g_a, g_b, g_c$ such that $\text{Re} \ g_a = a, \text{Re} \ g_b = b, \text{Re} \ g_c = c$ holds.
It's easy to note that $a = \text{Re} \dfrac{f}{|f|}$. Now, $$ \frac{f}{|f|} = \frac{f}{\sqrt{f \overline{f}}} = \sqrt{\frac{f}{\overline{f}}} $$
Similarly, we have $ b = \text{Re}\dfrac{f}{|f|^2}$, and $$ \frac{f}{|f|^2} = \frac{f}{f\overline{f}} = \frac{1}{\overline{f}}$$ or $c = \text{Re} \dfrac{f+if}{|f|^2}$, and $$ \frac{(1+i)f}{|f|^2} = \frac{(1+i)f}{f\overline{f}} = \frac{1+i}{\overline{f}},$$ but I'm not sure how to decide whether these functions are holomorphic or not...
I would appreciate any help. Thanks in advance!