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The problem now lies in finding the Green's function G that satisfies equation 1. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator L. Not every operator admits a Green's function. A Green's function can also be thought of as a right inverse of L.
Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation. There are many expansions in terms of special functions for the Green's function. In the case of a boundary put at infinity with the boundary condition ...
See Green's functions for the Laplacian or [2] for a detailed argument, with an alternative. It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function.
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely ...
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in R 3 {\displaystyle \mathbb {R} ^{3}} ).
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).
The Green's function to be used in the above integral is one which vanishes on the boundary: (,) = for and . Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
The Green's function, (~ ~ ′), for the d'Alembertian is defined by the equation (~ ~ ′) = (~ ~ ′)where (~ ~ ′) is the multidimensional Dirac delta function ...