# Thread: Harmonic functions and C-R equations

1. ## Harmonic functions and C-R equations

Please refer to the attached image.

Could I get some hints on how to evaluate these question.

The question asking to find where $\displaystyle f(re^{i \theta})$ is differentiable doesn't seem to involved,
however would I use C-R equations, or would it just be for wherever $\displaystyle r \neq 0$. Although that is given in the domain, so I'm assuming they want us to use C-R equations.
If i were to use C-R equations, then I would have to convert to cartesian coordinates, correct?
How would I do that.

As for the other question,
could I get any hints on what theorems may be relevant. I am not sure how to approach this.

Thanks

2. ## Re: Harmonic functions and C-R equations

Yes, use the Cauchy-Riemann Equations. You don't need to convert back to Cartesians, the Cauchy-Riemann Equations in polar form are \displaystyle \displaystyle \begin{align*} \frac{\partial u}{\partial r} = \frac{1}{r} \, \frac{\partial v}{\partial \theta} \end{align*} and \displaystyle \displaystyle \begin{align*} \frac{\partial v}{\partial r} = -\frac{1}{r}\,\frac{\partial u}{\partial \theta} \end{align*}.