# Green's Function

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## Green's function

A solution of a partial differential equation for the case of a point source of unit strength within the region under examination. The Green's function is an important mathematical tool that has application in many areas of theoretical physics including mechanics, electromagnetism, acoustics, solid-state physics, thermal physics, and the theory of elementary particles. The underlying physics in each of these areas is generally described by some linear partial differential equation which relates the physical variable of interest (electrostatic potential or pressure amplitude in a sound wave, for example) to a source function. For present purposes the source may be regarded as an independent entity, although in some applications (for example, particle physics) this view masks an inherent nonlinearity. The source may be physically located within the region of interest, it may be simulated by certain boundary conditions on the surface of that region, or it may consist of both possibilities. A Green's function is a solution to the relevant partial differential equation for the particular case of a point source of unit strength in the interior of the region and some designated boundary condition on the surface of the region. Solutions to the partial differential equation for a general source function and appropriate boundary condition can then be written in terms of certain volume and surface integrals of the Green's function.

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Green’s Function

a function related to the analytic representation of solutions of the boundary value problems of mathematical physics. In many cases, Green’s function makes possible a visual interpretation as a result of the action of a source of force or of a charge concentrated at a point. For this reason. Green’s function is sometimes called the function of a source. Thus, in an electrostatic interpretation, Green’s function represents the potential of the field of a point charge placed within a grounded conducting surface. It can be easily constructed for a number of regions (sphere, half-space, circle, rectangle, and so forth). The function is also used in solving boundary value problems for ordinary differential equations.

Green’s function plays an important role in theoretical physics, particularly in the quantum theory of fields and statistical physics. It describes the propagation of fields from the sources generating them (for this reason it is also called a function of propagation). The function was named after George Green, who first investigated a special case of it in his research on the theory of potential (1828).

### REFERENCES

Sobolev, S. L. Uravneniia matematicheskoi fiziki, 4th ed. Moscow, 1966.
Tikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.
Bogoliubov, N. N.. and D. V. Shirkov. Vvedenie v teoriiu kvan tovannykh polei. Moscow, 1957.
Mattuck. R. Feinmanovskie diagrammy v probleme mnogikh tel. Moscow. 1969. (Translated from English.)

## Green's function

[′grēnz ‚fəŋk·shən]
(mathematics)
A function, associated with a given boundary value problem, which appears as an integrand for an integral representation of the solution to the problem.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
The single-particle and collective excitations of the system under consideration manifest themselves as poles of the single-particle and two-particle Green's functions, but the corresponding expressions for the Green's functions cannot be evaluated exactly because the interaction part of the Hubbard Hamiltonian is quartic in the fermion fields.
The proposed embedding approach thus is significantly more efficient than computing the fields in the complete configuration by means of Green's functions of the empty casing [19, 20].
addresses an effective formulation of QCD which exhibits locality in fermionic Green's functions upon a sophisticated and highly nonperturbative functional integration (actually, functional differentiation) of gluonic gauge-field fluctuations, using the Halpern quadrature of the QCD partition function for the gauge field [A.sub.[mu]].
For a two-phase inhomogeneous material, the solution using the second derivatives of the Green's functions takes the form
It should be noted that the Green's functions play a vital role in the construction of an appropriate Fredholm integral equation.
In the next section, the discrete Green's functions and Huygens' principle for scalar waves and vector waves are formulated and derived.
Properties of Green's functions, which will be used in the main theorem, are given in the next theorem.
Generalized Green's functions and generalized inverses for linear differential systems with Stieltjes boundary conditions.
 have developed the steady-state Green's functions for forced vibrations of Timoshenko beam with damping effects.
Applications of Green's Functions in Science and Engineering (reprint, 1971)
Liu, "Simplified Green's functions for calculating capacitance and inductance of multiconductor transmission lines in multilayered media," IEEE Proceedings of Microwaves, Antennas and Propagation, Apr.1994, pp.

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