Instead of writing Gξ(x), we say that G(x, ξ) is aGreen’s function for the given boundary-value problem. The beauty of Green’s function is that it does not depend on the nonhomogeneous right side and, therefore, once we have found it, we automatically have the solution for any well-enough behaved f(x). (In fact, it turns out that one also can arrange to have the Green’s function reﬂect a whole class of boundary conditions. So, for example, we may set u(0) = α and u(L) = β and ﬁnd the corresponding Green’s function for arbitrary α and β.)
Of course, George Green lived long before any talk of the Dirac delta function and the like, so it seems that we should be able to “do” Green’s functions classically; we introduce them classically in this section. However, there is such a close connection between Green’s functions and the delta function that a discussion of one without the other would be misleading.