In this problem, we study one particular reprasentation of the Dirac delta-funct

Question

In this problem, we study one particular reprasentation of the Dirac delta-function is technically not a function but rather it refers to family of functions delta alpha(x) (indexed by alpha) that exhibit the following properties limalpharightarrow0 delta alpha(x) = 0 0x 0, limalpharightarrow0+infinity-infinity f(x)delta alpha(x)dx = f(0), for any continuous function f. Any such family provides a representation of the delta-function. When we use the delta-function in some formula, we are technically considering the limit behavior (as alpharightarrow0) generated by using the delta alpha in the formula. Here we will study one particular representation of the delta-function. For alpha > 0, consider the functions delta alpha(x) defined by delta alpha(x) : = 1/alpha0 for|x| alpha/2for |x| > alpha/2 (1) Prove that limalpharightarrow0 delta alpha(x) = 0 at all x 0. The mean value theorem for integration says that for any continuous function f. There exists some y (-alpha/2, alpha/2) such that alpha/2-alpha/2 f(x)dx = alpha f(y). Use this theorem to prove that limalpharightarrow0 +infinity-infinity f(x) delta alpha(x)dx = f(0) for every continuous function f. Consider the family of functions theta alpha(x) = 01/2 + x/2alpha1 if x alpha. Describe what the theta alpha(x) looks like as alpha rightarrow 0. What is the relationship between theta alpha(x) and delta alpha(x). Use the facts that delta(ax) = delta(x) if a > 0 and delta(ax) = 1 - delta(x) if a

Explanation / Answer

(x) = 1/ -/2 <= x < = /2

(x) = 0 x>/2 x< -/2

as -> 0

space for function where 1/ also tends to zero.

and function becomes -> 0 (x) = 0 for x not = 0 ;

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