Green's function

In physics and mathematics, Green's function is an auxiliary function in the solution of linear partial differential equations. The function is named for the British mathematician George Green (1793 – 1841)

Let Lx be a given linear differential operator in n variables x = (x1, x2, ..., xn), then the Green function of Lx is the function G(x,y) defined by

L_\mathbf{x} G(\mathbf{x},\mathbf{x}) = - \delta(\mathbf{x}- \mathbf{x}), $$ where &delta;(x-y) is the Dirac delta function. Once G(x,y) is known, any differential equation involving Lx is formally solved,

L_\mathbf{x} \,\phi(\mathbf{x}) = -\rho(\mathbf{x}) \quad\Longrightarrow\quad \phi(\mathbf{x}) = \int\; G(\mathbf{x},\mathbf{x})\; \rho(\mathbf{x})\; \mathrm{d}\mathbf{y}. $$ The proof is by verification,

L_\mathbf{x} \,\phi(\mathbf{x}) = \int\;L_\mathbf{x} \; G(\mathbf{x},\mathbf{x})\; \rho(\mathbf{x})\; \mathrm{d}\mathbf{y} = - \int\;\delta(\mathbf{x}- \mathbf{x}) \mathrm{d}\mathbf{y} = -\rho(\mathbf{x}) $$ where in the last step the defining property of the Dirac delta function is used.