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In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is causal whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one.
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 ...
The Green's function number specifies the coordinate system and the type of boundary conditions that a Green's function satisfies. The Green's function number has two parts, a letter designation followed by a number designation. The letter(s) designate the coordinate system, while the numbers designate the type of boundary conditions that are ...
is the derivative of the Green's function along the inward-pointing unit normal vector ^. The integration is performed on the boundary, with measure d s {\displaystyle ds} . The function ν ( s ) {\displaystyle \nu (s)} is given by the unique solution to the Fredholm integral equation of the second kind,
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 ...
The function K(x,y) is variously known as a Green's function, or the kernel of an integral. It is sometimes called the nucleus of the integral, whence the term nuclear operator arises. In the general theory, x and y may be points on any manifold ; the real number line or m -dimensional Euclidean space in the simplest cases.
If ω(λ) ≠ 0, the Green's function G λ (x,y) at λ is defined by (,) = {() / (), () / (). and is independent of the choice of Φ λ and Χ λ. In the examples there will be a third "bad" eigenfunction Ψ λ defined and holomorphic for λ not in [1, ∞) such that Ψ λ satisfies the boundary conditions at neither 0 nor ∞ .
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