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Find all real-valued $C^2$ differentiable functions $h$ defined on $(0,\infty)$ such that $u(x,y)=h(x^2+y^2)$ is harmonic on $\mathbb C-\{0\}$.

This is one of my homework problem. As I understand I should find a function defined on $(0,∞)$ satisfy the conditions above. So far I only got $u(x,y)=\operatorname{In}(x^2+y^2)$ and maybe we can expand this to $u(x,y)=\operatorname{In}(x^2+y^2)^n$ every $n$ except for $0$. so can you correct me if I am wrong at any point and also help me complete the solution.

Thanks everyone.

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1 Answer 1

up vote 2 down vote accepted

We have $$\partial_x [h(x^2+y^2)]=2x\cdot h'(x^2+y^2),$$ hence $$\partial_{xx}[h(x^2+y^2)]=2 h'(x^2+y^2)+4x^2h''(x^2+y^2).$$ Similarly, we obtain $$\partial_{yy}[h(x^2+y^2)]=2 h'(x^2+y^2)+4y^2h''(x^2+y^2).$$ The Laplacian of $u$ at $(x,y)$ can be expressed as a function of $x^2+y^2$, hence we get a differential equation that $h$ needs to satisfy on $(0,+\infty)$.

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let t=x^2+y^2 then i get h(t)=-log t. just to make sure since i got log(x^2+y^2). –  ruud Mar 10 at 21:29

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