Single-Particle Electrodynamics - Assassination Science

The adjective enumerated will be used rather than the adjective integral
where appropriate.
A.5 Special functions
A.5.1
Kronecker delta function
The Kronecker delta function δj i , taking enumerated arguments i and j, is
defined as
δ i
j =
{
1 if i = j
0 if i ≠ j.
(A.22)
A.5.2
Dirac delta function
The Dirac delta function δ(t), of real argument t, is defined as
δ(t) ≡ lim
ε→0
g(t, ε),
(A.23)
where g(t, ε) is any function in the ensemble of indefinitely differentiable
functions of real arguments t and ε > 0 such that g(t, ε) → 0 for all t ≠ 0 as
ε → 0, and
∫ ∞
−∞
dt ′ g(t ′ , ε) = 1.
This ensemble is referred to as the Dirac delta ensemble.
(A.24)
There is only one Dirac delta function in this thesis. This function may,
however, be instantiated for an arbitrary number of purposes. The limiting
procedure above accompanying its function definition is deemed to be executed
on the first blank page following the end of this thesis. If, at that point,
the evaluation of any given mathematical expression appearing in this thesis
is not invariant under a change of g(t, ε) through the Dirac delta ensemble,
then the expression in question cannot describe a physical quantity.
The three-dimensional Dirac delta function, of three-vector argument r,
is denoted δ (3) (r), or simply δ(r) where unambiguous, and is defined as
δ (3) (r) ≡ δ(x)δ(y)δ(z),
(A.25)
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