The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

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Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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Mean Value Property and Harmonic functions: a simple exercise

Let $f:[a,b]\to\mathbb{R}$ such that for every $h>0$ such that $$(x-h,x+h)\subset[a,b]$$ we have $$f(x)=\frac{1}{2}(f(x-h)+f(x+h)).$$ How can I conclude that $f$ is harmonic in $[a,b]$? My idea ...
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Evaluating a Limit with Generalized Harmonic Numbers.

Using WolframAlpha, I could informally come up with the following result: $$ \lim_{n \rightarrow \infty} \frac{H_n^{(-\frac{1}{2})}}{n\sqrt{n}} = \frac{2}{3} $$ Allowing me to infer that ...
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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? $$ H_{n}^{(r)} \in \mathcal{O}(?) $$
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Question about the maximum principle for the Laplace's equation

Maximum principle for the Laplace's equation: $$\nabla^2 u= \Delta u=u_{xx}+u_{yy}$$ $$\Delta u=f(x,y) \text{ Poisson }$$ Problem with boundary values of the form Dirichlet: $$\left.\begin{matrix} ...
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Prove that a function is harmonic by the use of the mean property

Given $y\in\mathbb{R}^N$, define $v_y:\mathbb{R}^N\setminus\{y\}\to\mathbb{R}$ by $$v_y(x) = \frac{|y|^2 - |x|^2}{|x-y|^N}.$$ By denoting $u(x) = |y|^2 - |x|^2$ and $w(x) = |y-x|^{-N}$, one can ...
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example of harmonic function on sphere

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren ...
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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) ...
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62 views

Harmonic conjugates on annulus slit

Let $D$ be an annulus slit with $$D= \{a<|z|< b \}$$ excluding $(-b,-a)$. Show that any harmonic function on $D$ has a harmonic conjugate on $D$. The hint says to fix $c$ between $a$ and ...
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52 views

What is a harmonic conjugate of $u=Arg(z)$?

Let $u=Arg(z)$ be a function maps $\mathbb{C}\setminus \{0\}$ to $ (-\pi,\pi]$. How do i find a harmonic conjugate of $u$ when $Arg(z)\in (-\pi,\pi)$?
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44 views

Extending bounded functions on $\mathbb{R}$ to $\mathbb{H}$ with the Poisson kernel.

Let $h(\phi):\mathbb{R}\to\mathbb{R}$ be a bounded piecewise continuous function on the real line. Define a function on the upper half-plane by the formula $\tilde{h}(s+it):=\int_{-\infty}^{\infty} ...
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42 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
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Looking at Green's Function for the Dirichlet Problem, how do you calculate n(y)

If G(x,y) = (-1/4pi)[(|x-y|^-1)-(|x-r(y)|^-1)] where r(y)=(y1,y2,-y3) and x and y are vectors in R3. The question states 'Evaluate ∂G(x,y)/∂ny (n subscript y) for y ∈ ∂Ω. I know to do this you ...
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Analytic property of harmonic functions through mean value property.

I have this strong intuition that one should be able to at least prove that a harmonic function is infinitely differentiable, let alone analytic, using just the mean value property. Unfortunately I am ...
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70 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
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Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
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Intuitive significance of harmonicity

I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it ...
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44 views

Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
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Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
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Chain Rule (applied twice) for vector valued functions

I need to show that if $f: \mathbb{R}^2 \to \mathbb{R}$ is harmonic, i.e., $$\frac{\partial^2 f}{\partial x^2}(x,y) + \frac{\partial^2 f}{\partial y^2}(x,y) = 0 \quad \text{ for all } (x,y) \in ...
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39 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
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Local regularity for harmonic functions (Laplace's equation)

I need a local Sobolev regularity result for a smooth solution $u$ of $$ -\Delta u=0 $$ with the equation satisfied in an open set $U$ (I have no boundary conditions). I know that such a smooth $u$ ...
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Second partial derivatives of harmonic functions

Given a twice-differentiable function $f:S\rightarrow \mathbb{R}$, where $S$ is a nonempty subset of $\mathbb{R}^n$, how does one prove that if f is harmonic then the second partials $D_jD_kf$ all ...
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An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
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45 views

Finding all harmonic functions satisfying initial conditions in the unit disk

Find all harmonic functions $\phi$ in the unit disk $D= \{\ z \in \mathbb{C} : |z|<1 \}\ $ that satisfy $\phi(\frac{1}{2})=4$ and $\phi(z)\ge 4$ for all $z \in D$. Through $\phi$ being harmonic, ...
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129 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 ...
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Harmonic function vanishing on a set of positive measure.

I'm preparing for a qualifying exam, and came across a question I couldn't figure out: If $\Omega$ is a region and $h:\Omega\to \mathbb{R}$ is a harmonic function vanishing on a set of positive ...
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Derivatives of the Green's Function

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. ...
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Laplace's equation, integral, tends to steady state?

If $v(x,y)$ solves Laplace's equation $v_{xx} + v_{yy} = 0$ on a bounded domain $S$, and $u(x,y,t)$ solves $u_t = u_{xx} + u_{yy}$ on $S$, with $u=v$ on $\partial S$ for all $t$, one can show that ...
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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 ...
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109 views

harmonic conjugate of a natural log

Find the harmonic conjugate of $\ln \sqrt{x^2+y^2}$ on some open nonempty subset of the plane. Ok I got stuck. So I set the function as $u(x,y)=\ln \sqrt{x^2+y^2}$. So to show that $u_x(x,y)= ...
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Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
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Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
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Harmonic functions on $\mathbb C-\{0\}$

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 ...
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Can we consider the points $(c,y)$ as local minima or maxima for all $y∈ℝ$

Let $f:(a,b)×ℝ→ℝ$ be a non zero and twice differentiable function. Let $c∈(a,b)$. Assume that $Δf=0$ (the Laplacian operator) and $f(c,y)=0$ for all $y∈ℝ$. Assume that $$f(x,y)<0=f(c,y)$$ for ...
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Conformal Mapping and Relating Solutions (of Laplace) of Domains (via the Mapping)

Find a conformal equivalence between the following domains: the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb ...
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The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
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Limit of natural log

Prove that $\displaystyle \lim_{n \to \infty} \ln x = \infty$ using the fact that the harmonic series diverges Of course, this is obvious graphically, but I have to prove it formally. I based my ...
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Harmonic Function With Step Function Boundary Data

Consider the Unit Disk. can we solve for a harmonic function in the unit disk such that: $\triangle u = 0 $ in D and $ u = f $ on $\partial D$ where $ f = 1$ for $|\theta| \leq \epsilon$ and $ |\theta ...
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66 views

Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
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Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
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Laplace's equation on a square domain with a central point reservoire

Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with ...
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Boundary integral of a harmonic function around a pole

I have a radial harmonic function $h:\mathbb R^N\backslash\{0\}\to\mathbb R$ which has a pole of order $m$ in 0, and I would like to compute $$ \frac{1}{\sigma_N}\int_{\partial ...
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Proof of strong maximum principle for harmonic functions

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 $x_0$, ...
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Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
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159 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
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Laplace's Equation in Spherical Coordinates

The general solution of the Laplace equation in spherical coordinates is (independant of $\phi$): $$V(r,\theta ) = \sum ^{\infty} _{l=0} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta ...
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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 ...
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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69 views

Function must be constant comparison

The following 2 problems are past exit exam problems for my major. I see that they're worded differently but are asking me to do the same thing. Not sure how they differ much I'd appreciate if anyone ...