Advanced Calculus fi..

526 Advanced Calculus, Fifth EditionFigure 7.18Function approximating the derivative S1(x).in the ordinary sense. A systematic development based on the ideas suggested here isgiven in the book by Lighthill listed at the end of the chapter. Here we simply list someof the results obtained and confine attention to generalized functions related to S(x).We find that generalized functions can be differentiated arbitrarily often andhence obtain Sr(x), 6"(x), .... The first derivative 6'(x) can be approximated bythe derivative of the pulse of Fig. 7.17; hence it appears as in Fig. 7.18. Similarapproximations are obtained for higher derivatives.A simple translation along the x-axis provides us with generalized functions ofform 6(x - c), 6'(x - c), ... for constant c.An ordinary function, say piecewise continuous, can also be regarded as a generalizedfunction. We confine attention here to ordinary functions that are formed ofa finite number of pieces having derivatives of all orders-for example, the functionf (x) = 0 for x < 0, = e-" for x 2 0. If we approximate f by a smooth functionf,(x), as we did for g(x) above, we are led to the conclusion that f (x) has a derivativefr(x) = 6(x) + h(x), where h(x) = 0 for x < 0 and h(x) = -e-" for x > 0 (thevalue at 0 being immaterial). Similarly, f "(x) is found to be 6'(x) - 6(x) + f (x).