Green's Function

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.

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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.

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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.)