ABSTRACT
EXAMPLE 10.1: DERIVATIVES OF DISCONTINUOUS FUNCTIONS Use the corresponding generalized functions to obtain the derivatives of all orders of the following ordinary functions that (1) are discontinuous; (2) have angular points, that is, discontinuous derivative; or (3) vanish outside a ”nite or in”nite interval, with a discontinuity of the function and/or of its derivative at one (two) end point(s). First, consider the functions’ negative exponential (hyperbolic sine) in the positive real half-axis, continued symmetrically to the negative real half-axis in Figure 10.1a (b), and obtain their derivates of all orders. Second, consider the functions’ circular cosine (sine) in Figure 10.1c (d) vanishing outside one period and obtain their derivatives of all orders. ¡ird, consider the functions’ hyperbolic cosine (sine) in Figure 10.1e (f) vanishing on the negative real axis and obtain their derivatives of all orders.
