Poisson equation using complex analytic method I have the following question in a complex analysis text: 
Find a particular solution to the following Poisson equation: 
$$\nabla^2u(r, \theta) = r^2 \cos \theta.$$ 
The solution method outlined in the text uses Wirtinger derivatives to simplify the equation, then integrate twice. So here we would have: 
$$4 \frac{\partial^2u}{\partial z \partial \bar{z}} = z \bar{z} \cos (\arg z).$$ 
Integrating with respect to $\bar{z}$ would then give: 
$$4 \frac{\partial u}{\partial z} = z \left( \frac{\bar{z}^2}{2} \right) \cos (\arg z) + f(z).$$ 
Now, here I'm stuck. I can't get this integration to work, in part because I'm not sure you can integrate $\arg(z)$. Any pointers on where I'm going wrong would be greatly appreciated. Thanks in advance. 

If $u$ is a solution to this one and $v$ is a solution to $\nabla^2 v(r,\theta) = r^2 \sin \theta)$, then $w = u + i v$ satisfies
$$ 4 \dfrac{\partial^2 w}{\partial z \partial \overline{z}} = \nabla^2 w = r^2 \exp(i\theta) = r z= z^{3/2} \overline{z}^{1/2}$$
Integrate with respect to $z$ and $\overline{z}$, and you find one solution is
$$ w  = \frac{z^{5/2} \overline{z}^{3/2}}{15} = \frac{r^{3} z}{15} = \frac{r^4}{15} \exp(i\theta)$$
So taking the real part, $$u = \frac{r^4}{15} \cos(\theta)$$
