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Find all real-valued functions $h$, defined and of class $C^2$ on the positive real line, such that the function $u(x,y)=h(x^2+y^2)$ is harmonic.

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Any of your own thoughts on the problem, etc. ? – muzzlator Apr 16 '13 at 17:13

We are looking for radial harmonic functions.

We have $u_x=2xh'(x^2+y^2)$ hence $u_{xx}=2h'(x^2+y^2)+4x^2h''(x^2+y^2)$, and in a similar way we can compute $u_{yy}$. This gives, letting $t:=x^2+y^2$, a differential equation satisfied by $h$.

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