Value Distribution of Biaxially Symmetric Harmonic Polynomials

1976 ◽  
Vol 28 (4) ◽  
pp. 769-773 ◽  
Author(s):  
J. D'Archangelo ◽  
P. A. McCoy
Keyword(s):  
Laplace Equation ◽  
Value Distribution ◽  
Harmonic Polynomials ◽  
Potential Equation ◽  
Symmetric Potential ◽  
Image Position

Consider the biaxially symmetric potential equationwhere α, β > — 1/2. If 2α + 1 and 2 β + 1 are non-negative integers and if X corresponds to the hypercirclethen the biaxisymmetric Laplace equation in E2(α+ β+2),and (1.1) are equivalent.

scholarly journals Nondegeneracy of the bubble for the critical p-Laplace equation

2020 ◽  
pp. 1-18
Author(s):  
Angela Pistoia ◽  
Giusi Vaira
Keyword(s):  
Laplace Equation ◽  
Sobolev Inequality ◽  
Quasilinear Equation ◽  
Image Position

Abstract We prove the non-degeneracy of the extremals of the Sobolev inequality \[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \] when 1 < p < N, as solutions of a critical quasilinear equation involving the p-Laplacian.

scholarly journals The iterated equation of generalized axially symmetric potential theory, VI General solutions

1974 ◽  
Vol 18 (3) ◽  
pp. 318-327
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Polar Coordinates ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Real Value ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory [1] is the equation where, in its simplest form, the operator Lk is defined by the function f f(x, y) being assumed to belong to the class of C2n functions and the parameter l to take any real value. In appropriate circumstances, which will be indicated later, the operator can be generalized but as this can be done without altering the methods used, the operator will be taken in the form where r, θ are polar coordinates such that x = r cos θ, y = r sin μ = cosθ.

scholarly journals Derivative-type ascent formulas for kernels of some half-space Dirichlet problems

2000 ◽  
Vol 42 (2) ◽  
pp. 185-194
Author(s):  
L. R. Bragg
Keyword(s):  
Potential Theory ◽  
Half Space ◽  
Laplace Equation ◽  
Bessel Functions ◽  
Axially Symmetric ◽  
Dirichlet Problems ◽  
Symmetric Potential ◽  
Derivative Type ◽  
Differentiation Formulas

AbstractDerivative-type ascent formulas are deduced for the kernels of certain half-space Dirichlet problems. These have the character of differentiation formulas for the Bessel functions but involve modifying variables after completing the differentiations. The Laplace equation and the equation of generalized axially-symmetric potential theory (GASPT) are considered in these. The methods employed also permit treating abstract versions of Dirichlet problems.

The lemma of the logarithmic derivative for subharmonic functions

1996 ◽  
Vol 120 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Lebesgue Measure ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Value Distribution Theory ◽  
Subharmonic Functions ◽  
Classical Statement ◽  
Image Position ◽  
Finite Lebesgue Measure

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

Polynomial Approximation and Growth of Generalized Axisymmetrig Potentials

1979 ◽  
Vol 31 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Peter A. McCoy
Keyword(s):  
Fourier Series ◽  
Initial Data ◽  
Polynomial Approximation ◽  
Jacobi Polynomials ◽  
Singular Line ◽  
Regular Solutions ◽  
Potential Equation ◽  
Ultraspherical Polynomials ◽  
Complete Set ◽  
Image Position

Generalized axisymmetric potentials Fα (GASP) are regular solutions to the generalized axisymmetric potential equation(1.1)in some neighborhood Ω of the origin where they are subject to the initial data(1.2)along the singular line y = 0. In Ω, these potentials may be uniquely expanded in terms of the complete set of normalized ultraspherical polynomials(1.3)defined from the symmetric Jacobi polynomials Pn(α, α)(ξ) of degree n with parameter α as Fourier series(1.4)

Electron trajectories in electrostatic fields

1992 ◽  
Vol 50 (2) ◽  
pp. 954-955
Author(s):  
Zbigniew Czyzewski ◽  
David C. Joy
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Magnetic Fields ◽  
Laplace Equation ◽  
Secondary Electron ◽  
Low Voltage ◽  
Electron Trajectories ◽  
Electrostatic Fields ◽  
Image Position ◽  
Detector Placement

Electron microscope use an electron beam to obtain various kind of information about specimen. The electron beam is focussed by electrostatic and magnetic fields and electron detectors employ electrostatic fields to attract or deflect electrons. In many cases the demand to calculate the electron trajectories in a fast and visual way is very strong. One of the most important questions is the problem of the secondary electron (SE) trajectories inside the SEM chamber and the effect of sample charging on detector yield. This is especially important in the low voltage SEM when investigating an uncoated, non-conductive specimen. A relatively large number of calculated trajectories gives a possibility to optimize SE detector placement as well as detector bias.The main problem is solving the Laplace equation in a 3-D space. In the 3-D space composed of cubic cells of dimension Δ3, the Laplace equation takes the following form:

The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

1970 ◽  
Vol 22 (4) ◽  
pp. 803-814 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Value Distribution ◽  
Minimum Modulus ◽  
Image Position

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

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