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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.
The unit disk is isomorphic to the upper half-plane by means of a Möbius transformation. For example, ... Let P r denote the Poisson kernel on the unit circle T.
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.
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group. Thus, the quotient of the upper half-plane by the action of the modular group is the so-called moduli space of elliptic curves: a space whose points describe isomorphism classes of elliptic curves. This is ...
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]
In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. It is one of the few stable distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.
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 Poisson wavelet family can be used to construct the family of Poisson wavelet transforms of functions defined the time domain. Since the Poisson wavelets satisfy the admissibility condition also, functions in the time domain can be reconstructed from their Poisson wavelet transforms using the formula for inverse continuous-time wavelet transforms.