The concept of higher order derivatives is useful in many branches of mathematics and its applications. As they are useful in many places, n^{th} order derivatives are often defined directly. **Higher Order Derivatives** discusses these derivatives, their uses, and the relations among them. It covers higher order generalized derivatives, including the Peano, d.l.V.P., and Abel derivatives; along with the symmetric and unsymmetric Riemann, Cesàro, Borel, L^{P}-, and Laplace derivatives.

Although much work has been done on the Peano and de la Vallée Poussin derivatives, there is a large amount of work to be done on the other higher order derivatives as their properties remain often virtually unexplored. This book introduces newcomers interested in the field of higher order derivatives to the present state of knowledge. Basic advanced real analysis is the only required background, and, although the special Denjoy integral has been used, knowledge of the Lebesgue integral should suffice.

Therefore, if k is even, then f(0) (x) exists and (s) f (x + t) + f (x − t) . 1 h→0 2 f(0) (x) = lim (1.6.2) Definition. A function f is called symmetrically continuous at x of even order if f (x + t) + f (x − t) = 2f (x) + o(1) as t → 0, (1.6.3) and symmetrically continuous at x of odd order if f (x + t) − f (x − t) = o(1) as t → 0. (1.6.4) In the literature, the condition (1.6.3) is used to mean that f is symmetric at x and condition (1.6.4) is used to mean that f is symmetrically

1 dt = o(h4 ) as h → 0. t2 (1.8.7) From (1.8.6) and (1.8.7), C2 Df (0) = 0. This shows that in Theorem 1.8.4 (ii) the inequalities may be strict and that for the last part of Corollary 1.8.5 the converse is not true. Example. Consider further the function g(x) = 2 sin 0, 2 1 4 1 1 − 2 cos 2 − 4 sin 2 , if x = 0, x2 x x x x if x = 0. Then, with f as in the previous example, f ′ (x) = g(x) for all x = 0. Since f is 40 Higher Order Derivatives continuous everywhere except x = 0 where it is

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(1 − r) lim (1 − r) δ r→1− f (t) δ 1 − r2 ∂ 2 2 ∂t2 1 ∆ dt = 0, we have πAS 2 f (0) = lim sup π(1−r) r→1− ∂2 f (r, x) ∂x2 δ x=0 = lim sup(1−r) r→1− 0 f (t)P ′′ (r, t) dt. 164 Higher Order Derivatives Therefore, the value of AS 2 f (0) does not depend on the values of f outside (0, δ). So, π π 1 − r2 dt > 0, for 0 < r < 1. ∆ −π 0 (2.17.37) n Since f (r, x) is given by (2.17.2), f (r, 0) = a0 /2 + ∞ a r and, thus, n=1 n f (r, 0) = −r 1 π f (t)P (r, t) dt = d d r f (r, 0) dr dr

this common value, possibly inﬁnite, is the right Peano derivative of f at x of order k + 1. + Note that the right upper derivate f (k+1) (x) deﬁned in section 4.1 is, in general, diﬀerent from Rf (k+1) (x) except for k = 0, since for k = 0 we are assuming the existence of Rf(k) (x) only and not f(k) (x) and since the existence of f(k) (x) implies that of Rf(k) (x) and not the converse. In a similar way one can deﬁne left derivatives and derivates that are denoted by Lf(k) (x) and Lf (k) (x), Lf