Let us find out more about the functions that bear Green's name. The analysis of modern formal definitions of the fundamental solution and Green's function demonstrates again how far George Green was ahead of his time.

The functions we call the fundamental solutions and Green's functions were not called functions until long after Green's time. In fact they are the so-called distributions. The "celebrity list" of distributions includes the Heaviside step function named after English engineer and mathematician Oliver Heaviside and the Dirac delta function named after English physicist Paul Dirac. Distributions have been "legitimized" only in 1935 by Sergei Sobolev and re-introduced in the late 1940s by Laurent Schwartz. To learn more about the distributions please visit the following sites:

• http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2010x-katz-cauchydirac.pdf

• http://www.amazon.com/Prehistory-Distributions-Mathematics-Physical-Sciences/dp/0387906479

• http://mathoverflow.net/questions/20314/good-books-on-theory-of-distributions

The fundamental solution is always related to a specific partial differential equation (PDE). For some equations it is possible to find the fundamental solutions from relatively simple arguments that do not directly involve "distributions." One such example is the Laplace's equation of the potential theory considered in Green's Essay.

Laplace's equation governs many important physical problems, e.g. electric and steady-state heat conduction, etc. The equation has a following form in the Cartesian coordinate system x, y, z.