Questions tagged [poisson-integrals]

For questions regarding Poisson integrals and Poisson kernels.

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Find the analytic function $f(z) = u(x,y) + iv(x,y)$ given that $f(z)$ is an integral.

I have the next problem: Find the analytic function $f(z) = u(x,y) + iv(x,y)$ given that: a) $ f(z)= \frac{1}{i\pi} \int_{ |\zeta|=R} \frac{u(\zeta)}{\zeta - z}d\zeta - \overline{f(0)}$ b) $u(z)= \...
2
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1answer
44 views

How can I justify that I can put the laplace operator under the integral?

Consider the integral: $ h(r, \theta) = \frac{1}{2\pi} \int_0^{2\pi} g(\phi) \frac{1-r^2}{1-2r \cos(\theta - \phi) + r^2} d\phi, r<1$ I want to show that $\Delta h=0$, but in order to do so, I need ...
2
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1answer
42 views

Relationship between Poisson's integral formula and the generating function of Chebyshev polynomials

On the disk $\{z:|z|<R\}$, Poisson's integral formula is $$u(r,\theta)=\frac1{2\pi}\int_0^{2\pi}\frac{(R^2-r^2)f(\phi)}{R^2-2Rr\cos(\theta-\phi)+r^2}\,d\phi$$ which solves the Dirichlet problem. ...
9
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1answer
90 views

Trouble with Poisson integral

I'm continuing my studies about the space $\mathbb{T}$ and I reach the point in which are introduced the Harmonic functions. Well up to now I have a little trouble with understanding the Poisson's ...
0
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1answer
85 views

Rado–Kneser–Choquet Theorem proof

I was reading the proof of the Rado–Kneser–Choquet Theorem. The statement is there in the image (taken from the book "Harmonic Mapping in the Plane, Duren page-$30$": In the proof, he shows that ...
0
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0answers
101 views

Poisson integral formula for boundary value problem

I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\...