# Tagged Questions

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 ...
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### Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
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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 ...
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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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### The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
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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 $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
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### Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of $u$...
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### Weakly Harmonic Functions (Weak Solutions to Laplace's Equation $\Delta u=0$) and Logic of Test Function Techniques.

In analysis we often use test functions $\phi\in C_{0}^{\infty}(U)$ in order to make some kind of deduction about another function $u:U\mapsto\mathbb{C}$. For example, if one can obtain the ...
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### Harmonic functions with zeros on two lines

For which pairs of lines $L_1$, $L_2$ do there exist real functions, harmonic in the whole plane, that are $0$ at all points of $L_1 \cup L_2$ without vanishing identically? Note: This is self-study ...
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### Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $\Bbb R^n \to \Bbb R^n$ with the property that, whenever $h$ is a harmonic function ($\Delta h=0$), $h\circ f$ is also harmonic. Is there a nice characterization of ...
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### A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
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### Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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### Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
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### Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
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Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) \... 1answer 78 views ### Is f(x) constant under these conditions? Statement Let f:\mathbb{R}\rightarrow \mathbb{R} be an function that is concave up and increasing. If \displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0, then f is constant. It'll be easy if ... 1answer 195 views ### Harmonic function in a unit disk with jump boundary data I am reading Conway's book about complex analysis. One question in it bothered me a lot recently. If given a piecewise continuous function with jump on the boundary of unit disk and it is bounded, we ... 1answer 560 views ### You can't solve Laplace's equation with boundary conditions on isolated points. But why? Consider a bounded region \Omega\subset\mathbb R^n with a finite number of "holes" X=\{x_1,\ldots,x_k\} that are isolated points in its interior. I'm pretty sure that in more than one dimension, ... 1answer 128 views ### Why is this function constant? I'm trying to prove that an entire harmonic function u that satisfies |u| \leq\sqrt{\sum_i |x_i|} is constant. I think that I have to use Liouville theorem for harmonic functions, however I don't ... 1answer 179 views ### Show \Omega is simply connected if every harmonic function has a conjugate Prove: If every harmonic function on \Omega has a harmonic conjugate on \Omega, then \Omega is simply connected. Same question is asked here but no proof is presented: is the converse true: in ... 1answer 65 views ### Prove that (\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2 Knowing: f(z) is analytical Prove:$$(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$$I have proved firstly that \ln|f(z)| is harmonic function Let$$...
Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...