Green's function

A Green's function is a kernel that represents the response of a linear differential operator to a point source. Integrating it against a source distribution yields a solution, subject to the operator's boundary conditions.

Dirac delta function: A generalized function that acts as a unit point source under integration. A Green's function is defined as the operator's response to this idealized source.
Distribution (mathematics): A continuous linear functional on a space of test functions, generalizing ordinary functions. Point sources and many Green's functions are naturally defined as distributions.
Poisson's equation: The elliptic equation relating a scalar field's Laplacian to a source distribution. Its Green's function computes potentials generated by charge or mass distributions.
Retarded Green's function: A Green's function that vanishes before a source acts, enforcing causal response. It selects the causal solution for time-dependent wave and field equations.
Linear differential operator: A linear map built from derivatives that acts on functions or distributions. The Green's function inverts this operator, subject to specified boundary conditions.
Impulse response: A system's output when its input is an impulse concentrated at one time or position. In linear systems, a Green's function is the differential-equation counterpart of an impulse response.
Helmholtz equation: A wave equation for a single frequency, involving the Laplacian and a wavenumber. Its Green's function describes the field radiated by a harmonic point source.
Advanced Green's function: A time-dependent Green's function that vanishes after a source acts. It contrasts with the causal, retarded choice by imposing a future-directed response condition.
Boundary-value problem: A differential equation whose solution must satisfy conditions at the domain's boundary. Boundary conditions determine which Green's function represents the desired solution.
Adjoint operator: An operator related to a given operator through an inner product or dual pairing. Adjoint Green's functions reveal reciprocity and help derive integral identities.

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