# Tagged Questions

For questions regarding harmonic functions.

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### Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
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### Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define ...
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### Holomorphic versus harmonic functions

Is it true that any holomorphic function on domain $D$ is of the form $u_x-iu_y$ for some harmonic function $u$? Motivation for this question is the problem of existence of harmonic conjugates.
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### Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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### Dirichlet problem on upper half disk

In our current Complex Variables homework assignment, we are given the following problem: Let $\Omega = \lbrace z \in \Bbb C : |z|<1, Im(z)>0\rbrace$ and $f \in C(\partial\Omega,\Bbb R)$ ...
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### Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
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### How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
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### Harmonic non-negative function is constant

I'm having some trouble with the following: Let $u:\mathbb{R}^2\setminus\{0\}\rightarrow[0,\infty)$ be a harmonic function. Show that $u$ is constant. I have seen different proves for this. However, ...
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### Neumann and Dirichlet Conditions for Schwarz-Christoffel Map

I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping. I'm able to map the polygon to the upper-half plane using ...
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### Proving $v$ is harmonic

Let $u$ be a harmonic function in $\mathbb{R}^3$ and let $a > 0$. Show that the function $v$ defined in spherical coordinates by $v(r,\theta,\psi )=\frac{a}{r}u(\frac{a^2}{r},\theta,\psi)$ is ...
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### All second partial derivatives of harmonic function are $0$

I am given this question as a homework assignment. Assume that $f$ is from $\mathbb R^2$ to $\mathbb R$ and has a strict local maximum at $(x_0, y_0)$. prove that all second partial derivatives of ...
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### Is there a conformal mapping that sends the upper semi-circle to the positive (or negative) real line,

and the real interval [-1,1] to the other half of the real line? I am considering the upper semi-circle ${|z|=1, 0<arg(z)<\pi}$, with the line [-1,1] that closes the loop. I want to map the ...
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### Showing an equation satisfies laplace equation

I'm completely lost with the laplace equation I've searched different explanations of it on google and on this website and nothing is helping explain it. The question I was given is: Show that the ...
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### Using Green's identity to show that a harmonic function with zero boundary values is identically zero

I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected. I have subbed in the information ...
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### How to find a family of analytic functions with a modulus of boundary value = 1, and no zeros inside the unit disk?

a) Find all $f(z)$ satisfying i) is analytic in $|z|\le 1$ ii) has no zeros in $|z|<1$ iii) $|f(e^{i\theta})|=1$ for $0<\theta<2\pi$ b) What are the possible $f(z)$ if (ii) is replaced by ...
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### Finding a harmonic function on half-disk that is equal to $1$ on the semi-circle and $0$ on the diameter

I first showed that the mapping $$z + \frac{1}{z}$$ sends the upper semi-disk, $\{|z|<1, \operatorname{Im} z >0\}$, along with the real line from $-1$ to $1$, to the whole of the real line in ...
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### Laplace's equation in cylindrical coordinates for a function that only depends on the angle

I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$ In cylindrical coordinates ($r,\theta,z$), for $\Phi =\Phi(\theta)$...
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Let $d=2$, and consider the domain $D=H$, the upper half-plane. Show that the Poisson kernel is the Cauchy distribution by following the steps below. (A) Let $W_t=(X_t,Y_t)$. Show that for any $\... 1answer 54 views ### Laplace equation problem: difficulty finding a particular solution given the boundary conditions. Solve Laplace's equation: $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$$ for the region$\{(x, y) : 0 < x < 1, 0 < y < 1\}\$, given these boundary conditions: ...
Say after finding a conformal mapping, using this conformal mapping, I find a harmonic function $$\arg \left[\frac{(z^2+1)}{(z^2-1)}\right]$$ that satisfies boundary conditions that I want. Is it ...