Green's function
In
mathematics, if
L is a
linear operator acting upon distributions over a
manifold,
M, then any solution of (
Lf)(
x) = δ(
x −
x0), where δ is the
Dirac delta function, is called a
Green's function of
L at
x0. If the
kernel of L is nontrivial, then the Green's function is not unique. However, in practice, some combination of
symmetry, boundary conditions and/or other externally imposed criteria would give us a unique Green's function. Also, please note Green's functions are distributions in general, not functions.
Not every operator L admits a Green's function, though. A Green's function can also be thought of as a one-sided inverse of L.
Motivation
Convolving with a Green's function gives solutions to inhomogeneous differentio-integral equations. If g is the Green's function of an operator L, then the solution for f of the equation Lf = h is the convolution of g with h.
Examples
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