Editing Green's function (section)

==Definition and uses==
A Green's function, {{math|''G''(''x'',''s'')}}, of a linear [[differential operator]] {{math|1=''L'' = ''L''(''x'')}} acting on [[distribution (mathematics)|distributions]] over a subset of the [[Euclidean space]] {{nowrap|<math>\R^n</math>,}} at a point {{mvar|s}}, is any solution of
{{NumBlk|1=|2=<math display="block">L\,G(x,s) = \delta(s-x) \, ,</math>|3={{EquationRef|1}}}}
where {{mvar|δ}} is the [[Dirac delta function]]. This property of a Green's function can be exploited to solve differential equations of the form
{{NumBlk|1=|2=<math display="block">L\,u(x) = f(x) \,.</math>|3={{EquationRef|2}}}}

If the [[kernel (linear operator)|kernel]] of {{math|''L''}} is non-trivial, then the Green's function is not unique. However, in practice, some combination of [[symmetry]], [[boundary condition]]s and/or other externally imposed criteria will give a unique Green's function. Green's functions may be categorized, by the type of boundary conditions satisfied, by a [[Green's function number]]. Also, Green's functions in general are [[Distribution (mathematics)|distributions]], not necessarily [[Function (mathematics)|functions]] of a real variable.

Green's functions are also useful tools in solving [[wave equation]]s and [[diffusion equation]]s. In [[quantum mechanics]], Green's function of the [[Hamiltonian mechanics|Hamiltonian]] is a key concept with important links to the concept of [[density of states]].

The Green's function as used in physics is usually defined with the opposite sign, instead. That is,
<math display="block">L G(x,s) = \delta(x-s)\,.</math>
This definition does not significantly change any of the properties of Green's function due to the evenness of the Dirac delta function.

If the operator is [[translation invariant]], that is, when <math>L</math> has [[constant coefficients]] with respect to {{mvar|x}}, then the Green's function can be taken to be a [[convolution kernel]], that is,
<math display="block">G(x,s) = G(x-s)\,.</math>
In this case, Green's function is the same as the impulse response of [[LTI system theory|linear time-invariant system theory]].