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.

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).


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]
A function, associated with a given boundary value problem, which appears as an integrand for an integral representation of the solution to the problem.
References in periodicals archive ?
Let us underline, that the causality condition for Green's function is equivalent to the requirement of analyticity of its transient Fourier transformation in theupper half-plane of the complex frequencies + .
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Amendola, "Analysis of substrate integrated waveguide structures based on the parallel-plate waveguide Green's function," IEEE Transactions on Microwave Theory and Techniques, Vol.
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where k is the wave number ([omega] [square root of ([micro][epsilon])]), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the magnetic vector potential Green's function,
Before proceeding with the approximation of Green's function we will remind that a probabilistic impulse response characteristics of a single unit charge in a free 3D space at the time moment t is
In Section 4 we use the asymptotic formula to obtain that the covariance of the height fluctuation is given by the Green's function (2).
Thus, the problem of finding a particular solution for a given boundary value problem, expressed as an ordinary differential equation, will be reduced to finding the associated Green's function.
In the following part we are going to investigate the properties of the core K(M, M', t) and compare them with the properties of the theoretical core expressed by the Green's function.
Empirical Green's function study of the January 17, 1994 Northridge, California earthquake.
This model couples a fracture-mechanics source model and a Rayleigh-wave Green's function.