# 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, $$u = \frac{r^4}{15} \cos(\theta)$$
• Got it. Thanks very much. Sorry about the nitpicking but it should be $\nabla^2 v(r, \theta)$ in the top line, right? – K.Reeves Jul 8 '17 at 3:31
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Note that $\ds{\cos\pars{\theta} = {x \over r} = {z + \bar{z} \over 2\root{\vphantom{\large a^{a}}z\bar{z}}}\quad}$ and $\ds{\quad\mrm{u}\pars{r,\theta} = \mrm{U}\pars{r\expo{\ic\theta},r\expo{-\ic\theta}}}$.