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In mathematics, a Green's function is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions. Via the superposition principle, the convolution of a Green's function with an arbitrary function ''f''(''x'') on that domain is the solution to the inhomogeneous differential equation for ''f''(''x''). In other words, given a linear ODE, , we can first solve , for each s, and realizing that since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L. Green's functions are named after the British mathematician George Green, who first developed the concept in the 1830s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In Quantum field theory, Green's functions take the roles of propagators. ==Definition and uses== A Green's function, ''G''(''x'', ''s''), of a linear differential operator ''L'' = ''L''(''x'') acting on distributions over a subset of the Euclidean space R''n'', at a point ''s'', is any solution of where is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form If the kernel of ''L'' is non-trivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function. Also, Green's functions in general are distributions, not necessarily proper functions. Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, the Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. As a side note, the Green's function as used in physics is usually defined with the opposite sign; that is, : This definition does not significantly change any of the properties of the Green's function. If the operator is translation invariant, that is, when ''L'' has constant coefficients with respect to ''x'', then the Green's function can be taken to be a convolution operator, that is, : In this case, the Green's function is the same as the impulse response of linear time-invariant system theory. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Green's function」の詳細全文を読む スポンサード リンク
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