Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $(0,+\infty)$.

share|cite|improve this answer
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 '14 at 21:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.