Boggio's formula

Last updated September 07, 2019

In the mathematical field of potential theory, Boggio's formula is an explicit formula for the Green's function for the polyharmonic Dirichlet problem on the ball of radius 1. It was discovered by the Italian mathematician Tommaso Boggio.

In mathematics and mathematical physics, potential theory is the study of harmonic functions.

In mathematics, a Green's function of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions is its impulse response.

In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

The polyharmonic problem is to find a function u satisfying

(-\Delta )^{m}u(x)=f(x)

where m is a positive integer, and $(-\Delta )$ represents the Laplace operator. The Green's function is a function satisfying

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols $\nabla\cdot\nabla$ , $\nabla 2$ . The Laplacian $\nabla\cdot\nabla f (p)$ of a function $f$ at a point $p$ , is the rate at which the average value of $f$ over spheres centered at $p$ deviates from $f (p)$ as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

(-\Delta )^{m}G(x,y)=\delta (x-y)

where $\delta$ represents the Dirac delta distribution, and in addition is equal to 0 up to order m-1 at the boundary.

In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

Boggio found that the Green's function on the ball in n spatial dimensions is

G_{m,n}(x,y)=C_{m,n}|x-y|^{2m-n}\int _{1}^{\frac {\left||x|y-{\frac {x}{|x|}}\right|}{|x-y|}}(v^{2}-1)^{m-1}v^{1-n}dv

The constant $C_{m,n}$ is given by

C_{m,n}={\frac {1}{ne_{n}4^{m-1}((m-1)!)^{2}}},

where

e_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma (1+{\frac {n}{2}})}}

Sources

Boggio, Tomas (1905), "Sulle funzioni di Green d'ordine m" (PDF), Rendiconti del Circolo Matematico di Palermo , 20, pp. 97–135, doi:10.1007/BF03014033
Gazzola, Filippo; Grunau, Hans-Christoph; Sweers, Guido (2010), Polyharmonic Boundary Value Problems (PDF), Lecture Notes in Mathematics, 1991, Berlin: Springer, ISBN 978-3-642-12244-6

In computing, a digital object identifier (DOI) is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

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