# Tagged Questions

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### Prove u=0 for a harmonic function u.

Let $\Omega$ be an open connected set in $\mathbb{R}^n$ with the boundary $\partial\Omega$ of class $C^2$. Let $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ be a harmonic function in $\Omega$, such ...
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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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### implicit derivates incorperating laplace's equation

If $f(x,y)$ is a harmonic function show that the function $F(x,y)=f(x^2-y^2,2xy)$ is also harmonic. You have to use Laplace's formula to prove this, unless there is an easier way. I'm having trouble ...
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### Harmonic Conjugate in Star Domain

I have been given that $u(x,y)$ is a harmonic function on a star shaped domain $D$. I have to show that it has harmonic conjugate $v(x,y)$ on same domain given up to additive constant by ...
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### Harmonic function in the upper half plane

Find an harmonic function $h(z)$ in the upper half plane with the following properties: $h$ is bounded in $\mathbb{C}_+$ and is continuous in $\{z | \Im(z) \geq 0 \}\setminus \{0 \}$ ...
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### 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. ...
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### Show that a harmonic function $u(r, \theta)$ dependent only on $r$ has the form $u(r, \theta) = a \log r + b$

What I have done so far is this: I've shown that if $u(r, \theta)$ is a harmonic function dependent only on $r$ then Laplace's equation becomes $u_{rr} + \frac{1}{r}u_r = 0$ I've also shown ...
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### Why is the harmonic function $\log(x^2 + y^2)$ not the real part of any function that is analytic in $\mathbb{C} - \{0\}$?

I would like to show that $\log(x^2 + y^2)$ is not the real part of any analytic function in $\mathbb{C} - \{0\}$ A similar question can be found here, but I don't think this argument is ...
### Showing that if $u$ is a real-valued harmonic function then for any real $c$ we have that $u^{-1}(c)$ is unbounded
I have the following homework question: Let $u$ be a non-constant real-valued harmonic function in $\mathbb{C}$. Prove that $u^{-1}(c)$ is unbounded for every real number $c$ There is a hint ...