2
votes
1answer
67 views

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 ...
1
vote
0answers
284 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
0
votes
0answers
98 views

Green first identity and harmonic function

I proved the first Green identity $$\int_{\partial\Omega}f\cdot(D_ng)d\partial\Omega=\int_\Omega \bigtriangledown f \cdot \bigtriangledown g + f\bigtriangledown ^2g d\Omega$$ and now I need to prove ...
3
votes
1answer
115 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
3
votes
0answers
763 views

Finding the harmonic conjugate

I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the ...