Can anyone tell me how to calculate the Poisson kernel for the upper half plane? I am able to calculate it for the unit disc and I know the unit disc and the upper half plane are conformally equivalent, do I need this?
Poisson kernel for upper half plane
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complex-analysis
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1You probably don't need this, but I'd probably try to use the [Cayley-transform](http://en.wikipedia.org/wiki/Cayley_transform#Conformal_map) to guess the Poisson kernel for the upper half plane. If you get the formula right, then it should be possible to derive it without resorting to the unit disk, but I don't quite see the point (and I haven't checked). *Added:* Actually [Wikipedia seems to answer this question](http://en.wikipedia.org/wiki/Poisson_kernel#On_the_upper_half-plane). Could you please try and be a bit more precise what exactly you're asking? I must say that I'm a bit confused. – 2011-05-12
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This is a good exercise. Let $\phi$ be the conformal mapping of the half plane to the unit disk.
To create a harmonic function on $\mathbb{H}$ which agrees with $f$ on the real line, one good strategy would be to translate it to the unit disk. Using the Poisson kernel for the disk, we can find a harmonic function on the disk which agrees with $f\circ \phi^{-1}$ on the boundary. Compose it with $\phi$ (which is also harmonic) to get a function which is harmonic on $\mathbb{H}$ that agrees with $f$ on the real line.
This is an outline, in the sense that to derive the Poisson kernel for the upper half plane, you have to power through some algebraic manipulations. That is messy, but not hard (especially if you know the answer).
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0@theo: actually i am trying to calculate harmonic measure for upper half plane to do that i need to calculate that poison kernel and in a book I got poisson kernel for upper half plane is 1by pie multiplication with something.I found the detail of finding poisson kernel of unit and then calculated its harmonic measure.so I need to learn how to calculate for upperhalf plane in a bit detail. thank you – 2011-05-12
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0Sorry. I lost you. Above is a method for calculating the Poisson kernel for the upper half plane. Do you need to calculate the harmonic measure? – 2011-05-12
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0yes i want to calculate harmonic measure for upper half plane – 2011-05-15
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0Given that you know how to calculate the Poisson kernel $p(z,t), z\in \mathbb{H}, t\in \mathbb{R}$for the upper half plane, the harmonic measure of an interval $I\subset \mathbb{R}$ w.r.t $z\in \mathbb{H}$ is given by the intergral of $p(z,t)$ over $I$. So one thing: you are calculating harmonic measure of a measurable subset of the boundary, NOT in the interior. – 2012-10-06
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0To Har : I have a question regarding your answer: I agree with you that, switching to the disk model $D$, you can find a harmonic function $F$ on $D$ which agrees with $f\circ \phi^{-1}$. But if you post compose $F$ with the holomorphic $\phi$, why is that harmonic again ? Pre-composition of harmonic with holomorphic produces a harmonic function, but not the post-composition. Actually, this is a question I ran into recently : Let $f$ be continuous, strictly increasing map from real line to itself, let $f^$ be the corresponding map in $S^1$. Then what is the connection between ... – 2012-10-06
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0...the harmonic extension of $f$ in $\mathbb{H}$ and the harmonic extension of $f^$ in $\mathbb{D}$ ? I would have been happier if they would have just been the same, but I am not sure that's true ! – 2012-10-06