Problem 2.24 Delta functions live under integral signs, and two expressions (D₁ (x)and D₂(x)) involving delta functions are said to be equal if[ + f(x)D₁(x) dx = [+0° f(x)D₂(x) dx.-8-00for every (ordinary) function f(x).(a) Show that8(cx) =-178(x).[2.142]where c is a real constant. (Be sure to check the case where c is negative.)(b) Let 0(x) be the step function:{0(x)=1.[2.143](In the rare case where it actually matters, we define (0) to be 1/2.) Showthat de/dx = 8(x).if x > 0.if x < 0.0.

The expressions involving delta functions D1x and D2x are said to be equal if,…

Problem 2.24 Delta functions live under integral signs, and two expressions (D₁(x) and D₂(x)) involving delta functions are said to be equal if +∞ +00 [ f(x)D₁(x) dx = [ f(x)D₂(x) dx. for every (ordinary) function f(x). (a) Show that 8(cx) = -8(x) |c| [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let (x) be the step function: 1. if x > 0. 0. if x < 0. 0(x)= [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x).

Problem 2.24 Delta functions live under integral signs, and two expressions (D₁(x) and D₂(x)) involving delta functions are said to be equal if +∞ +00 [ f(x)D₁(x) dx = [ f(x)D₂(x) dx. for every (ordinary) function f(x). (a) Show that 8(cx) = -8(x) |c| [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let (x) be the step function: 1. if x > 0. 0. if x < 0. 0(x)= [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x).