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

$L_{\mathbf {x} }G(\mathbf {x} ,\mathbf {x} )=-\delta (\mathbf {x} -\mathbf {x} ),$

where δ(x-y) is the Dirac delta function. Once G(x,y) is known, any differential equation involving L_x 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.