Advanced Calculus fi..

Advanced Calculus, Fifth EditionA simple example is that of density, say for mass distributed along a line, thex-axis. If p(x) is the density (in units of mass per unit length), then lab p(x)dxshould give the total mass in the interval a 5 x 5 b. Now suppose that all our massis concentrated in a single particle of mass 1, located at the origin. It is temptingto assign a density p(x) such that p(x) dx = 0 if the interval a 5 x 5 b doesJabnot contain the origin and such that la p(x) dx = 1 if the interval does contain theorigin. However, these conditions would make p(x) = 0 except "near the origin,"where p(x) must suddenly become infinite so rapidly that the integral of p(x) overeach small interval containing the origin is 1. This particular density "function" isthe Dirac delta function 6(x). Thus we require:The apparently contradictory nature of these properties led to their rejection asmeaningless by mathematicians until recently. Then a number of mathematicians,notably L. Schwartz, developed systematic theories of generalized functions, whichsatisfy all the desired properties in a rigorous manner. The function 6(x) is such ageneralized function.We can arrive at the delta function in another way, which suggests a limit processfor obtaining this and other generalized functions. Let g(b) denote the total mass onthe interval -m < x 5 b, for a distribution of mass (with finite total mass) on thex-axis. Thus for a continuous density p(x), g(b) = fb; p(x) dx and gl(b) = p(b).Now for the case of a single particle of mass 1 at the origin we have difficulty indefining p(x) but can easily find g(x) for each x. For g(x) is simply the total massto the left of x; hence for the case of the single particle,g(x) = 0 for x < 0, g(x) = 1 for x > 0. (7.107)Thus g(x) has a jump discontinuity at x = 0. We call the function g(x) in (7.107)the Heaviside unit function.Now from this g(x) we can again try to obtain the density p(x) as the derivativeof g(x). It is clear that gf(x) = 0 for x # 0, and it is reasonable to write g'(0) = oo,because of the jump. Thus we havegl(x) = 0 = S(x) for x # 0, gf(0) = oo = S(0). (7.108)Accordingly, we interpret the "function" 6(x) as the derivative ofthe Heaviside unitfunction. Now the unit function really has no derivative at x = 0, so we have notcompletely clarified the meaning of 6(x). To go further, we approximate g(x) by asmooth function g,(x) having a derivative for all x; we choose an 6 > 0 and set1 1g,(x) = 1 - -e-'/' for x 2 0, g,(x) = -ex/' for x < 0 (7.109)2 2(see Fig. 7.16). The function g, (x) has been chosen so that as 6 + 0+, g, (x) + g(x)(except at x = 0, where g,(x) = for all 6). Furthermore, g,(x) has a derivative1 -x/c 1g: (x) = -e for x 2 0, g:(x) = -ex'' for x < 0, (7.110)26 26