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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).

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