Editing Green's function (section)

==Green's functions for solving non-homogeneous boundary value problems==
The primary use of Green's functions in mathematics is to solve non-homogeneous [[boundary value problem]]s. In modern [[theoretical physics]], Green's functions are also usually used as [[propagator]]s in [[Feynman diagram]]s; the term ''Green's function'' is often further used for any [[correlation function (quantum field theory)|correlation function]].

===Framework===
Let <math>L</math> be the [[Sturm–Liouville theory|Sturm–Liouville]] operator, a linear differential operator of the form
<math display="block">L = \dfrac{d}{dx} \left[p(x) \dfrac{d}{dx}\right] + q(x)</math>
and let <math>\mathbf{D}</math> be the vector-valued [[boundary condition]]s operator
<math display="block">\mathbf{D} u = \begin{bmatrix}
\alpha_1 u'(0)    + \beta_1 u(0) \\
\alpha_2 u'(\ell) + \beta_2 u(\ell)
\end{bmatrix} \,.</math>

Let <math>f(x)</math> be a [[continuous function]] in {{nowrap|<math>[0,\ell]\,</math>.}} Further suppose that the problem
<math display="block">\begin{align}
 Lu &= f \\
 \mathbf{D}u &= \mathbf{0}
\end{align}</math>
is "regular", i.e., the only solution for <math>f(x) = 0</math> for all {{mvar|x}} is{{nowrap| <math>u(x) = 0</math>.}}{{efn|In technical jargon "regular" means that only the [[Trivial (mathematics)|trivial]] solution {{nowrap|(<math>u(x) = 0</math>)}} exists for the [[homogenization (mathematics)|homogeneous]] problem {{nowrap|(<math>f(x) = 0</math>).}}}}

===Theorem===
There is one and only one solution <math>u(x)</math> that satisfies
<math display="block"> \begin{align}
 Lu & = f \\
 \mathbf{D}u & = \mathbf{0}
\end{align}</math>
and it is given by
<math display="block">u(x) = \int_0^\ell f(s) \, G(x,s) \, ds\,,</math>
where <math>G(x,s)</math> is a Green's function satisfying the following conditions:
# <math>G(x,s)</math> is continuous in <math>x</math> and <math>s</math>.
# For {{nowrap|<math>x \ne s\,</math>,}} {{pad|4em}} {{nowrap|<math> L G(x,s) = 0</math>.}}
# For {{nowrap|<math>s \ne 0\,</math>,}} {{pad|4em}} {{nowrap|<math> \mathbf{D}G(x,s) = \mathbf{0}</math>.}}
# [[Derivative]] "jump": {{pad|0.5em}} {{nowrap|<math> G'(s_{0+},s) - G'(s_{0-},s) = 1 / p(s) \, </math>.}}
# Symmetry: {{pad|4em}} {{nowrap|<math> G(x,s) = G(s,x) \,</math>.}}

===Advanced and retarded Green's functions===
{{See also|Green's function (many-body theory)|propagator}}
Green's function is not necessarily unique since the addition of any solution of the homogeneous equation to one Green's function results in another Green's function. Therefore if the homogeneous equation has nontrivial solutions, multiple Green's functions exist. In some cases, it is possible to find one Green's function that is nonvanishing only for <math>s \leq x</math>, which is called a retarded Green's function, and another Green's function that is nonvanishing only for <math>s \geq x </math>, which is called an advanced Green's function. In such cases, any linear combination of the two Green's functions is also a valid Green's function. The terminology advanced and retarded is especially useful when the variable x corresponds to time. In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is [[causal]] whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one. The use of advanced and retarded Green's function is especially common for the analysis of solutions of the [[inhomogeneous electromagnetic wave equation]].