This problem concerns the following approximation to the delta function:

\f(a) Find the function C'() such that [: dz 5. (z) = 1. If x has units of length, what are the units of ,(x)? (b) Show that B =1 e0 0 otherwise. lim ,(x) = { () Compute Show that x> 0. \fHere the integral is defined as f " dz6(2) (@) = lim [ " d28,(2) (), o0 where the limit 0 is taken after evaluating the integral. Here (x) is a smooth test function that goes to zero rapidly as x +o0. (e) Use the result from part (d) to show that as a statement about distributions. This means that this relation holds when inte- grated against an arbitrary test function