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 ...
7
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1answer
1k views

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 ...
6
votes
3answers
3k views

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 $g$ ...
5
votes
1answer
438 views

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 ...
2
votes
1answer
730 views

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 $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
3
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1answer
209 views

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 ...
17
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2answers
1k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
9
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2answers
498 views

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 ...
8
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1answer
2k views

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 ...
3
votes
3answers
147 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
9
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1answer
501 views

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 ...
4
votes
2answers
565 views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad ...
4
votes
1answer
135 views

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 ...
3
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1answer
80 views

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 ...
3
votes
1answer
187 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
7
votes
3answers
291 views

Bounded, non-constant harmonic functions: how far are they from existing?

Let $f$ be a function that maps $\mathbb{Z}^2$ to $\mathbb{R}$ and consider the operator $T$ which replaces the value of $f$ at $(i,j)$ by the average of the values of $f$ at its four neighbors: $$ ...
3
votes
1answer
254 views

Two question on harmonic function

In a question paper I got the following two questions. $u(z)=u(x,y)$ be a harmonic function in $\mathbb{C}$ satisfying $u(z)\le a|\ln|z||+b$ for some positive constants $a,b$ and for all complex ...
1
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1answer
2k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
0
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0answers
95 views

Determining the Asymptotic Order of Growth of the Generalized Harmonic Function?

How should I proceed to determine the order of growth of the generalized harmonic numbers? $$ H_{n}^{(r)} \in \mathcal{O}(?) $$
0
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1answer
129 views

Choose parameters to make a harmonic function

Let $B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ ...
0
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2answers
959 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.
8
votes
1answer
363 views

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, ...
5
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1answer
183 views

the real part of a holomorphic function on C \ {0, 1}

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω = \text{C \ {0, 1}}$. Show that there exist unique real numbers $a_0$, $a_1$ such that $u(z) = h(z) − a_0 \log |z| − a_1 \log ...
3
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0answers
37 views

Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
5
votes
1answer
411 views

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 $u \in C(\Omega)$ ...
5
votes
1answer
656 views

Laplace equation Dirichlet problem on punctured unit ball.

Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering ...
3
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1answer
42 views

Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ ...
3
votes
2answers
237 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 ...
3
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0answers
96 views

Maps in $\mathbb{R}^n$ preserving harmonic functions

A map $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ preserves harmonic functions if $f\circ\varphi$ is harmonic for every harmonic function $f:\mathbb{R}^n\to\mathbb{R}$. It is known that these maps are, in ...
3
votes
1answer
215 views

Laplace's Equation in Polar Coordinates

I am trying to express Laplace's equation in terms of polar coordinates. That is, $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\\ x=r\cos\theta,\\ y=r\sin\theta. $$ My book ...
0
votes
2answers
935 views

Derive the Poisson Formula for a bounded C-harmonic function in the upper half-plane.

My book gives the Poisson Formula for such a harmonic function as: $$ u(x + iy) = \frac{1}{\pi} \int_{-\infty}^{\infty}{\frac{y \cdot u(t) dt}{(t - x)^2 + y^2}} $$ Here is what I have attempted. ...
5
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2answers
105 views

Is this Harmonic Polynomial Identically Zero?

Let $h$ be a harmonic polynomial that is zero on the lines $Im(z) = 1$ and $Im(z) = -1$. I know that by the Schwarz Reflection Principle, $h$ must be zero on any line $Im(z) = k$ for $k$ odd, but ...
4
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1answer
1k views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
3
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1answer
59 views

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 ...
3
votes
1answer
252 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
2
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1answer
127 views

Laplacian Boundary Value Problem

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f ...
1
vote
1answer
384 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
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vote
2answers
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Show that the Poisson kernel is harmonic as a function in x over $B_1(0)\setminus\left\{0\right\}$

Show that the Poisson-kernel $$ P(x,\xi):=\frac{1-\lVert x\rVert^2}{\lVert x-\xi\rVert^n}\text{ for }x\in B_1(0)\subset\mathbb{R}^n, \xi\in S_1(0) $$ is harmonic as a function in $x$ on ...
0
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1answer
525 views

Proof that laplace's equation is rotationally invariant using chain rule

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) ...
4
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1answer
525 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, ...
3
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1answer
64 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 ...
3
votes
1answer
172 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 ...
3
votes
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 ...
2
votes
0answers
127 views

$u(x,y)$ harmonic and bounded in punctured disc; show $0$ is a removable singularity [duplicate]

I'm working on a problem from p. 166 of Lars Ahlfors' Complex Analysis: If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity in the sense that ...
1
vote
1answer
36 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
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1answer
55 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
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2answers
62 views

Laplace $2$-D Heat Conduction

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space ...
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vote
1answer
88 views

Function $u(x,y) \in C^2$ that admits harmonic conjugate

Problem 1) Prove that if the real and imaginary part of a holomorphic function are of class $C^2$, then they are harmonic. 2)Deduce from 1) that if $u(x,y) \in C^2$ is a function that admits a ...
1
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1answer
565 views

Constants in Laplace's equation for a cube

I'm working a Laplace's equation $\Delta F=0$ for a cube in Cartesian coordinates $((0,0,0),(a,a,a))$ and after separation I have $$\frac{X''(x)}{X(x)} +\frac{Y''(y)}{Y(y)}+\frac{Z''(z)}{Z(z)} = ...
1
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1answer
191 views

Harmonic Function Transformation Help

Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...