Second-order partial differential equations and Greens functions

Second-Order Partial Differential Equations and Green s Functions 

The function 5(x — x0) is especially useful and was popularised by Dirac. There are many ways to represent its behaviour, one of which is 

Alternatively, a Gaussian could have been used [491, 499]. In the limit A -+ 0, only when x is very close to x0 (i-e- x — x0 is about zero) does the delta function depart from zero and there it tends to infinity. From the definition of the delta function [e.g. eqn. (310)] 

In three dimensions, the delta function S(r — r0) is only non-zero when the components of r, (x, y, z) are simultaneously very close to the components of r0, (x0,yu,zo) i-e  

Inhomogeneous second-order (partial) differential equations 

---