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"Find all harmonic functions $f:\mathbb{C}\backslash\{0\} \to \mathbb{R}$ that are constant on every circle centered at 0." This is one of the past qualifying exam problems that I was working on.

I was thinking to deal with $\frac{1}{f}$ so that $\frac{1}{f}$ is defined at 0 and use Schwarz lemma or something like that.

Any help or guidance would be really appreciated.

Thank you in advance.

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    It is Schwarz b$y$ the way, not Schwartz.2012-12-30

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First of all, you should check (this is a good exercise in using the chain rule) that the Laplace operator in polar coordinates is given by $\Delta f = \frac{1}{r}\frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}.$

If $f$ is constant on circles centered at $0$, then $f$ depends only on $r$ (i.e. derivatives with respect to $\theta$ are $0$), so $f$ must satisfy $\frac{1}{r}\frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} = 0.$ This equation can be viewed as a first order, ordinary differential equation in $u = \frac{\partial f}{\partial r}$: $u' + \frac{1}{r} u = 0,$ whose solutions are $u = \frac{C}{r}.$ Hence,$f = C\log r + D = C\log|z| + D.$

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Here's a hint, which I think will allow you to solve it:

You can represent $f(z)$ in polar form $f(z)=f(R,\phi)$ with $R\in\mathbb{R}$ and $z=Re^{i\phi}$ ($\phi$ is only defined up to $2\pi$, of course). Then write the Cauchy-Riemann equations for $R$ and $\phi$ (they're available here). In this form, you know that $\partial f/\partial \phi=0$, for every $R$.