Editing Green's function (section)

===Combining Green's functions===
If the differential operator <math>L</math> can be factored as <math>L = L_1 L_2</math> then the Green's function of <math>L</math> can be constructed from the Green's functions for <math>L_1</math> and {{nowrap|<math>L_2</math>:}}
<math display="block"> G(x, s) = \int G_2(x, s_1) \, G_1(s_1, s) \, ds_1. </math>
The above identity follows immediately from taking <math>G(x, s)</math> to be the representation of the right operator inverse of {{nowrap|<math>L</math>,}} analogous to how for the [[Invertible matrix#Other properties|invertible linear operator]] {{nowrap|<math>C</math>,}} defined by {{nowrap|<math>C = (AB)^{-1} = B^{-1} A^{-1}</math>,}} is represented by its matrix elements {{nowrap|<math>C_{i,j}</math>.}}

A further identity follows for differential operators that are scalar polynomials of the derivative, {{nowrap|<math>L = P_N(\partial_x)</math>.}} The [[fundamental theorem of algebra]], combined with the fact that <math>\partial_x</math> [[Commutative property|commutes with itself]], guarantees that the polynomial can be factored, putting <math>L</math> in the form:
<math display="block"> L = \prod_{i=1}^N \left(\partial_x - z_i\right),</math>
where <math>z_i</math> are the zeros of {{nowrap|<math>P_N(z)</math>.}} Taking the [[Fourier transform]] of <math>L G(x, s) = \delta(x - s)</math> with respect to both <math>x</math> and <math>s</math> gives:
<math display="block"> \widehat{G}(k_x, k_s) = \frac{\delta(k_x - k_s)}{\prod_{i=1}^N (ik_x - z_i)}. </math>
The fraction can then be split into a sum using a [[partial fraction decomposition]] before Fourier transforming back to <math>x</math> and <math>s</math> space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if <math>L = \left(\partial_x + \gamma\right) \left(\partial_x + \alpha\right)^2</math> then one form for its Green's function is:
<math display="block"> \begin{align}
G(x, s) & = \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\gamma(x-s)} - \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\alpha(x-s)} + \frac{1}{\gamma-\alpha} \Theta(x - s) \left(x - s\right) e^{-\alpha(x-s)} \\[1ex]
& = \int \Theta(x - s_1) \left(x - s_1\right) e^{-\alpha(x-s_1)} \Theta(s_1 - s) e^{-\gamma (s_1 - s)} \, ds_1.
\end{align} </math>
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when <math>\nabla^2</math> is the operator in the polynomial).