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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.
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 ...
Cayley transform of upper complex half-plane to unit disk. On the upper half of the complex plane, the Cayley transform is: [1] [2] = +.Since {,,} is mapped to {,,}, and Möbius transformations permute the generalised circles in the complex plane, maps the real line to the unit circle.
A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
Integral contour for deriving Kramers–Kronig relations. The proof begins with an application of Cauchy's residue theorem for complex integration. Given any analytic function in the closed upper half-plane, the function ′ (′) / (′), where is real, is analytic in the (open) upper half-plane.
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 ).
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 ...
The path C is the concatenation of the paths C 1 and C 2.. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = e i a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z 1, z 2, …, z n.