By I. G. Aramanovich, R. S. Guter, L. A. Lyusternik and I. N. Sneddon (Auth.)

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H2 + ··· hn + OQt*1). 42) THE DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE 29 If in a neighbourhood of the point x = x0 e (a, b) the function f(x) is represented in the form f(x0 + h) = a0 + ath + a2h2 + · · · + anhn + o(hn), then this expansion is unique, there exist at the point x = x0,n first differential derivatives (see §2) and the coefficients at (i = 0, 1, 2 , . . ,ri)satisfy the relationships βι = / ' ( * > ) , a0=f(xo), a2 = Ö ^ - , . . , 2! 42) are called Taylor's formulae. In particular, when x0 = 0 they are also called Maclaurin's formulae.

20) and in general, f o r / e Ck dkf(X, h) = ( f A, A ) / W .

56) 42 MATHEMATICAL ANALYSIS n (b) A generalization of Leibniz 's formula. If P„(t) = £ ciktk9 then fc = 0 n Pn(D)(Uiu2) = £ atDHuM) = t^(^>)"i]"2 + [ΛΧε>Κ]Ζ>κ2 fc=0 + i - \p"n{p)uxWu2 + ^ - [p;"(/))Wl]i)3W2 + ... 57) 72! or, which is shorter, n 1 Pn(P) (UlU2) = Σ — [ W ^ l t f ^ , Λ=Ο A:! 58) where P™(t) denotes the &th derivative of the polynomial Pn(t). \X) + - + ρ<"°(λ)} ^ . EXAMPLE 16. 59) P'i'(/) = 6, PI3V(0 = 0. Therefore P3{D)ekx = (^L - 2 — + l\e** = (λ3 - 2λ + l)e**. dx ) \dx3 In the same way P3(D) ( Λ ^ s f ^ l - 2 ~ + l ) (*****) V&3