Finding Harmonic conjugate for $\arg(z)$ 
Let $u(z)=\arg(z)$ in $D=\mathbb{C} \setminus\mathbb{R}^-$ where $\mathbb{R}^-=(-\infty,0]$. Find Harmonic conjugate for $u$.

Any ideas / hints?

I think I might have a simpler solution that requires absolutely no knowledge of complex logarithms. 
If we use polar coordinates, then we get $u(r,\theta)=\theta$. Now, we want to find a function $v(r,\theta)$ so that $u(r,\theta) + iv(r,\theta)$ will be holomorphic. Meaning, we have to make sure $u$ and $v$ will satisfy the polar version of the Cauchy-Riemann equations: $$\frac{\partial u}{\partial r}=\frac 1r\frac{\partial v}{\partial \theta},\frac{\partial v}{\partial r}=-\frac 1r\frac{\partial u}{\partial \theta}$$
In our case these translate into the following: $$\frac{\partial v}{\partial \theta}=0,\frac{\partial v}{\partial r}=-\frac 1r$$
This means that $v$ is only a function of $r$ which we can solve for using simple single-real-variable integration to get: $$v(r,\theta)=-\ln r$$
