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The unit disk may be conformally mapped to the upper half-plane by means of certain Möbius transformations.Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane.
is the fundamental solution of the Laplace equation in the upper half-plane. [59] It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. [60]
The "slit realisation" as the unit disk as the extended complex plane with [−1,1] removed comes from the mapping z = (w + w −1)/2. [22] On the other hand the map (z + 1)/(z − 1) carries the extended plane with [−1,1] removed onto the complex plane with (−∞,0] removed. Taking the principal value of the square root gives a conformal ...
For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if f ( z ) is analytic in the upper half complex plane { z : Im{ z } > 0} , and u ( t ) = Re{ f ( t + 0· i )} , then Im{ f ( t + 0· i )} = H( u )( t ) up to an ...
In complex analysis, the Hardy spaces (or Hardy classes) are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz ( Riesz 1923 ), who named them after G. H. Hardy , because of the paper ( Hardy 1915 ).
The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle.; The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.
If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H 2, the Poincaré half-plane model, and PSL(2, R) is the group of all orientation-preserving isometries of H 2 in this model.