I recently learnt that every harmonic function occurs as the real part of a complex analytic function. We also know that every harmonic function is real analytic. So, when is a real-analytic function harmonic ?

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    $\begingroup$ What kind of answer are you looking for? I suppose "$u$ is harmonic when $\Delta u = 0$" is not what you're after. $\endgroup$ – mrf Jun 16 '14 at 7:24
  • $\begingroup$ I want a criterion to say that a given power series with real co-efficients is harmonic. We know that every power series is not harmonic.I hope I'm not missing something fundamental here . $\endgroup$ – Srinivas K Jun 16 '14 at 7:27

When $-\Delta u=-\sum_{k=1}^n\frac{\partial^2u}{\partial x_k^2}=0$

I believe an equivalent property is the mean integral property, I.e.:

$u(x)=\frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)}u(y)dS(y)=\frac{1}{\alpha(n)r^{n}}\int_{ B(x,r)}u(y)dy$, $\forall r\gt 0$.

Where $\alpha(n)$ is the volume on the unit sphere in $\Bbb R^n$.


A real valued function can never be analytic except a constant function.let it be a $~f(z)=x~$ so it have to satisfy CR equation. but here $~Ux=1~$ but $~Vy=0~$. So CR equation not satisfied so not differentiable any where.so we can't talk about analytic but it is hormonic.

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    $\begingroup$ An analytic function is a function that has a convergent power series. The only complex-analytic function that is real valued is a constant function, but there are many real analytic functions that are not constant; e.g. $e^{-|x|^2}$. $\endgroup$ – robjohn Jul 24 at 19:13
  • $\begingroup$ Your function is not analytic.Reconfirm it please $\endgroup$ – Rishi Kushwaha Jul 25 at 14:52
  • $\begingroup$ It is real analytic on $\mathbb{R}^n$. $\endgroup$ – robjohn Jul 25 at 15:03
  • $\begingroup$ but I'm talking in complex $\endgroup$ – Rishi Kushwaha Aug 1 at 16:19
  • $\begingroup$ The question specifically says "real-analytic", so complex analytic functions are not what the question is about. $\endgroup$ – robjohn Aug 1 at 17:10

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