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I was looking at a comprehensive exam, and I found the this question. Can anyone help me out?

If $u$ is a harmonic function, which type of function $f$ is needed so that $f(u)$ is harmonic?

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    Maybe [this](http://math.stackexchange.com/q/85513/19341) might be interesting...2012-03-30

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Note that $u$ is harmonic if and only if $\Delta u:=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0.$ Therefore, if $u$ is harmonic, by chain rule we have $\Delta f(u)=\frac{df}{du}\Big(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\Big)+\frac{d^2f}{du^2}\Big(\big(\frac{\partial u}{\partial x}\big)^2+\big(\frac{\partial u}{\partial y}\big)^2\Big)=\frac{d^2f}{du^2}\Big(\big(\frac{\partial u}{\partial x}\big)^2+\big(\frac{\partial u}{\partial y}\big)^2\Big).$ If $u$ is a constant function (which is harmonic), then $\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=0$. It follows from the above formula that $f(u)$ is harmonic for any function $f$. On the other hand, if $u$ is a nonconstant harmonic function, then $\big(\frac{\partial u}{\partial x}\big)^2+\big(\frac{\partial u}{\partial y}\big)^2\neq 0$. Again it follows from the above formula that $f(u)$ is harmonic when $\frac{d^2f}{du^2}=0$, that is, when $f$ is a linear function in $u$: $f(u)=Au+B,$ where $A$ and $B$ are constants.

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    Yes, you are right. That was a typo. I corrected it.2011-11-13