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# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$

My problem is to integrate this expression: $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$. I know the answer is zero. Can you explain you idea to me or just prove ...
7k views

### Composition of a harmonic function with a holomorphic function is still harmonic

If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ is a conformal mapping of a domain $D_0$ onto $D$, is $f \circ g$ harmonic in $D_0$? I noticed this question while reading ...
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### If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity

This is a reworking of a previous question here which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the ...
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### Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
729 views

### Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where $C>0$...
3k views

### Positive harmonic function on $\mathbb{R}^n$ is a constant?

Is it true that a positive harmonic function on $\mathbb{R}^n$ must be a constant? How might we show this? The mean value property seems not to be the way...for that we would need boundedness.
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### Maximum principle for subharmonic functions

Let $\Omega$ be a domain of $\mathbb{R}^n$, and $u:\Omega\to\mathbb{R}$ a continuous function. We call $u$ subharmonic if for any ball $B\subset\subset\Omega$ and any $h:\overline B\to\mathbb{R}$ ...
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### How to prove Liouville's theorem for subharmonic functions

I noticed this post and this paper, which gives a version of Liouville's theorem for subharmonic functions and the reference of its proof, but I think there must be an easier proof for the following ...
547 views

### If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set $U$...
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### Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
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### Show that the Kelvin-transform is harmonic

First, I have to give you our definitions: Consider $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. The function $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
380 views

### Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $z \in \mathbb{C}$. Prove that $u$ is constant. [duplicate]

Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $z \in \mathbb{C}$. Prove that $u$ is constant. I think i should use Liouville's theorem, but how can i ...
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### is there a way to prove the Mean-value formulas using complex analysis?

THEOREM : $U$ is an open subset of $\mathbb{R^n}$ and suppose $u \in C^2(U)$ is harmonic within $U$, then : u(x)= \def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{B(x,r)}u\,dy = \def\...
### $f$ is analytic, nonzero on simply connected domain. Show that $\log|f(z)|$ is harmonic.
$f$ is analytic, nonzero on a simply connected domain $\Omega \subset \Bbb C$. Show that $\log|f(z)|$ is harmonic on $\Omega$. I thought of two methods: $f=u+iv$ so $\log|f(z)|=\log\sqrt{u^2+v^2}$ ,...
### $f \in C^2(\mathbb{R}^2)$ is harmonic; $f$ is in one $L^p(\mathbb{R}^2)$ $\implies$ $f$ is constant
Suppose $f \in C^2(\mathbb{R}^2)$ is harmonic. Suppose that $f$ is in one $L^p(\mathbb{R}^2)$ space, with $1 \le p \le \infty$. How can I prove that $f$ is constant and in particular is identically \$...