Editing Green's function (section)

{{short description|Impulse response of an inhomogeneous linear differential operator}}
{{about|the classical approach to Green's functions|a modern discussion|fundamental solution}}
{{short description|Non-linear second-order differential equation}}
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[[File:Green's function animation.gif|alt=An animation that shows how Green's functions can be superposed to solve a differential equation subject to an arbitrary source.|thumb|360x360px|If one knows the solution <math display="inline">G(x,x')</math> to a differential equation subject to a point source <math display="inline">\hat{L}(x) G(x,x') = \delta(x-x')</math> and the differential operator <math display="inline">\hat{L}(x)</math> is linear, then one can superpose them to build the solution <math display="inline">u(x) = \int f(x') G(x,x') \, dx'</math> for a general source <math display="inline">\hat{L}(x) u(x) = f(x)</math>.]]
In [[mathematics]], a '''Green's function''' (or '''Green function'''<ref>{{Cite journal |last=Wright |first=M. C. M. |date=2006-10-01 |title=Green function or Green's function? |url=https://www.nature.com/articles/nphys411 |journal=Nature Physics |language=en |volume=2 |issue=10 |pages=646–646 |doi=10.1038/nphys411 |issn=1745-2473}}</ref>) is the [[impulse response]] of an [[inhomogeneous ordinary differential equation|inhomogeneous]] linear [[differential operator]] defined on a domain with specified initial conditions or boundary conditions.

This means that if <math>L</math> is a linear differential operator, then

* the Green's function <math>G</math> is the solution of the equation {{nowrap|<math>L G = \delta</math>,}} where <math>\delta</math> is [[Dirac delta function|Dirac's delta function]];
* the solution of the initial-value problem <math>L y = f</math> is the [[convolution]] {{nowrap|(<math>G \ast f</math>).}}

Through the [[superposition principle]], given a [[linear differential equation|linear ordinary differential equation]] (ODE), {{nowrap|<math>L y = f</math>,}} one can first solve {{nowrap|<math>L G = \delta_s</math>,}} for each {{mvar|s}}, and realizing that, since the source is a sum of [[delta function]]s, the solution is a sum of Green's functions as well, by linearity of {{mvar|L}}.

Green's functions are named after the British [[mathematician]] [[George Green (mathematician)|George Green]], who first developed the concept in the 1820s. In the modern study of linear [[partial differential equation]]s, Green's functions are studied largely from the point of view of [[fundamental solution]]s instead.

Under [[Green's function (many-body theory)|many-body theory]], the term is also used in [[physics]], specifically in [[quantum field theory]], [[aerodynamics]], [[aeroacoustics]], [[electrodynamics]], [[seismology]] and [[statistical field theory]], to refer to various types of [[correlation function (quantum field theory)|correlation functions]], even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of [[propagator]]s.