## Transcribed Text

Q 11. Properties of the Dirac delta “function”
Consider the generalized integrator function δ(x) , defined by its proper-
ties:
ε −ε
∞ −∞
dxδ(x)=1x∗= 1 ifx∗≥0, dxδ(x−c)f(x)=f(c), ∀c∈R,
x∗ −∞
dxδ(x) = 1, ∀ε > 0,
0 ifx∗<0
where dxδ(x) is understood as a slight abuse of notation and f(x) in the last formula is a suitably well-behaved (at least bounded and continuous – and perhaps even smoother – in a neighborhood of x=c) function of x. Remember that δ(x) can be defined by a limiting process with respect to a sequence of functions chosen by our desire/need for smoothness vs. compactness (∼ localization to some bounded interval of x) in any particular instance, e.g.:
δ(x) = lim n 1x − 1x− 1 n→∞ n
2 1 =limn x+
0 ifx<0orx≥n1 n if 0 ≤ x < 1
n
−nx2/2
2πt Hint: consider the change of variables y ≡ αx.
= lim
, ,
n→∞
1x+1 −2x1x+ x− 1x−1
1 n→∞nn nn
n n→∞ 2π
e−x2/2t = lim √ .
= lim
(a) Express δ(αx) for α ∈ R in terms of δ(x). What is δ(−x)?
Write δ(g(x)) and ∞
integral
−∞
∞ −∞
e
t→0
(b) Consider a bounded C1 function g(x) with a finite set of real roots: {x∗i ∈R:g(x∗i)=0,i=1,...,M},
where all M of the x∗i are assumed to be simple roots, i.e. g′(x=x∗i ) ̸= 0.
∗
(c) Express δ(x−c) and δ(α(x−c)) for α ∈ R and x, c ∈ RN in terms of the N individual components δ(xi−ci),i = 1,...,N.
dx δ(g(x))f (x) in terms of the xi . Interpret the value of the dx δ(g(x)) |g′(x)| in relation to the properties of g(x).
′ ′ δ(x+h)−δ(x) (d) Define the derivative δ (x) of δ(x) by: δ (x) = lim .
h→0 h
Assuming that the usual integration by parts formula dg f = f g − df g holds for
g(x)=δ′(x) and g(x)=δ(x), evaluate
∞ −∞
dx δ′(x − c)f(x) for f(x) ∈ C1(R).

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.