In mathematics, a Green's 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 ${\displaystyle \operatorname {L} }$ is the linear differential operator, then

• the Green's function ${\displaystyle G}$ is the solution of the equation ${\displaystyle \operatorname {L} G=\delta }$, where ${\displaystyle \delta }$ is Dirac's delta function;
• the solution of the initial-value problem ${\displaystyle \operatorname {L} y=f}$ is the convolution (${\displaystyle G\ast f}$).

Through the superposition principle, given a linear ordinary differential equation (ODE), ${\displaystyle \operatorname {L} y=f}$, one can first solve ${\displaystyle \operatorname {L} G=\delta _{s}}$, 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 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.

Under 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 functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.