D.2 Some Green’s functions

D.2.1 The Poisson equation

The so-called Poisson equation

is

Here

The solution Green’s function

Loosely speaking, the above integral solution chops function

Note that often, the Poisson equation is written without a minus sign.
Then there will be a minus sign in

The objective is now to derive the above Green’s function. To do so, first an intuitive derivation will be given and then a more rigorous one. (See also chapter 13.3.4 for a more physical derivation in terms of electrostatics.)

The intuitive derivation defines delta function,

located at the
origin. That means that

Here

By itself the above definition is of course meaningless: infinity is
not a valid number. To give it meaning, it is necessary to define an
approximate delta function, one that is merely a large spike rather
than an infinite one. This approximate delta function

In the above integral the region of integration should at least include the small region of radius

The corresponding approximate Green’s function

In the limit

To find the approximate Green’s function, it will be assumed that

Now integrate both sides of the Poisson equation over a sphere of a
chosen radius

As noted, the delta function integrates to 1 as long as the vicinity of the origin is included. That means that the right hand side is 1 as long as

According to the [divergence] [Gauss] [Ostrogradsky] theorem, the left hand side can be written as a surface integral to give

Here

Because

The exact Green’s function

Finally the rigorous derivation without using poorly defined things
like delta functions. In the supposed general solution of the Poisson
equation given earlier, change integration variable to

It is to be shown that the function

Here

It will be assumed that the function

In particular, this approximation becomes exact in the limits where the constants

as can be verified by explicitly differentiating out the three terms of the integrand. Next note that the third term is zero, because as seen above

D.2.2 The screened Poisson equation

The so-called screened Poisson equation

is

Here

The analysis of the screened Poisson equation is almost the same as
for the Poisson equation given in the previous subsection. Therefore
only the differences will be noted here. The approximate Green’s
function must satisfy, away from the origin,

The solution to this that vanishes at infinity is of the form

where

over a sphere of radius

The rigorous derivation is the same as before save for an additional