Green's function

In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

This means that if ${\displaystyle L}$ is a linear differential operator, then

• the Green's function ${\displaystyle G}$ is the solution of the equation ${\displaystyle LG=\delta ,}$ where ${\displaystyle \delta }$ is Dirac's delta function;
• the solution of the inhomogeneous problem ${\displaystyle Ly=f}$ is the convolution, ${\displaystyle y=(G\ast f).}$

Through the superposition principle, given a linear ordinary differential equation (ODE), ${\displaystyle Ly=f}$, one can first solve ${\displaystyle LG=\delta _{s}}$, for each s. If the source is a sum of delta functions, then the solution is a sum of Green's functions as well due to linearity of L. This means that the integral, viewed as a continuous sum, can reconstruct a wide class of sources, ${\displaystyle f}$, through the convolution integral. Whenever the integral of ${\displaystyle f}$ with ${\displaystyle G}$ converges, then the solution to the inhomogeneous equation, ${\displaystyle Ly=f}$, is given by y = (${\displaystyle G\ast f}$).

Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead, which take into account the modern language of Distribution theory.

Building off of the superposition principle in many-body theory, the term is also used in physics and engineering, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology 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 role of propagators, also referred to as two-point (correlation) functions.