Editing Green's function (section)

==Further examples==
* Let {{math|1=''n'' = 1}} and let the subset be all of {{math|'''R'''}}. Let {{mvar|L}} be <math display="inline">\frac{d}{dx}</math>. Then, the [[Heaviside step function]] {{math|Θ(''x'' − ''x''<sub>0</sub>)}} is a Green's function of {{math|''L''}} at {{math|''x''<sub>0</sub>}}.
* Let {{math|1=''n'' = 2}} and let the subset be the quarter-plane {{math|1={(''x'', ''y'') : ''x'', ''y'' ≥ 0}<nowiki/>}} and {{mvar|L}} be the [[Laplacian]]. Also, assume a [[Dirichlet boundary condition]] is imposed at {{math|1=''x'' = 0}} and a [[Neumann boundary condition]] is imposed at {{math|1=''y'' = 0}}. Then the X10Y20 Green's function is <math display="block"> \begin{align}
G(x, y, x_0, y_0) =
\dfrac{1}{2\pi}
&\left[\ln\sqrt{\left(x-x_0\right)^2+\left(y-y_0\right)^2} - \ln\sqrt{\left(x+x_0\right)^2 + \left(y-y_0\right)^2} \right. \\[5pt]
&\left. {} + \ln\sqrt{\left(x-x_0\right)^2 + \left(y+y_0\right)^2}- \ln\sqrt{\left(x+x_0\right)^2 + \left(y+y_0\right)^2} \, \right].
\end{align}</math>
* Let <math> a < x < b </math>, and all three are elements of the real numbers. Then, for any function <math>f:\mathbb{R}\to\mathbb{R}</math> with an <math>n</math>-th derivative that is integrable over the interval <math>[a, b]</math>: <math display="block">
f(x) = \sum_{m=0}^{n-1} \frac{(x - a)^m}{m!} \left[ \frac{d^m f}{d x^m} \right]_{x=a} + \int_a^b \left[\frac{(x - s)^{n-1}}{(n-1)!} \Theta(x - s)\right] \left[ \frac{d^n f}{dx^n} \right]_{x=s} ds \,.</math> The Green's function in the above equation, <math>G(x,s) = \frac{(x - s)^{n-1}}{(n-1)!} \Theta(x - s)</math>, is not unique. How is the equation modified if <math>g(x-s)</math> is added to <math>G(x,s)</math>, where <math>g(x)</math> satisfies <math display="inline">\frac{d^n g}{d x^n} = 0</math> for all <math>x \in [a, b]</math> (for example, <math>g(x) = -x/2</math> with {{nowrap|<math>n=2</math>)?}} Also, compare the above equation to the form of a [[Taylor series]] centered at <math>x = a</math>.