On the Poisson Representation of a Function Harmonic in the Upper Half-Plane
Copyright policy )On the Poisson Representation of a Function Harmonic in the Upper Half-Plane
Let \(R_q (z,t)=\frac 1{\pi} \operatorname{Im} \frac{(1+tz)^q}{(t-z)(1+t^2)^q}\), where \(q =0,1,2, \dots\). The authors propose the representation \(u(z)=\int_{-\infty}^{+\infty} R_q (z,t) \,d\nu(t)+ \operatorname{Im} P_q(z)\) for a function \(u(z)\) harmonic in the upper half-plane with a polynomial \(P_q(z)\) under some restriction on the Borel measure \(\nu\). For \(q=0\) this representation becomes the classic Poisson formula for the upper half-plane.
Phragmén-Lindelöf principle, Boundary value and inverse problems for harmonic functions in two dimensions, Phragmen - Lindel of Principle, Compactness, Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions, Nevanlinna formula, Poisson formula, Green function, Subharmonic function, Integral representations, integral operators, integral equations methods in two dimensions, Poisson integral, Harmonic, subharmonic, superharmonic functions in two dimensions, integral representation of harmonic function
Phragmén-Lindelöf principle, Boundary value and inverse problems for harmonic functions in two dimensions, Phragmen - Lindel of Principle, Compactness, Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions, Nevanlinna formula, Poisson formula, Green function, Subharmonic function, Integral representations, integral operators, integral equations methods in two dimensions, Poisson integral, Harmonic, subharmonic, superharmonic functions in two dimensions, integral representation of harmonic function
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