By Jordan C.
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Extra resources for Calculus of Finite Differences
Since the expansion of u,(f) is simpler than that of u(f), we will use the former for the determination of f(x). We get u, (f) zz f”’ Finally we have i f x57. , (-1)” [ :,“I fY = 9, [ m+zy-‘] tmtv. , 22 To determine the generating functions there are several methods. First method. In Infinitesimal Calculus a great number of expansions in power series are known. Each function expanded may be considered as a generating function. For instance we found: 1. 1 s f” l--t= a=0 2. e’ = So -$ f” . , G$=e’ 3.
By the first divided difference of f(xi), denoted by Ff (xi) the following quantity is understood: Zf (xi) = f (xi:! Q+ -j- fo x1-x0 ’ W(4 = (x f(X”l ! fk) 0 --x,1 h---x,) -i- (x’--xJ (XI--XJ + f(x2) + (q--x,) (x2-q ’ I and so on. Putting W,,,(X) = (x-x,,) (x-x,) . , . pf(x,) = &k-g + -w-- -f . . ,. fkol X0 x0 m-l Xl2 . . 1 Xl fkl Xl . , . , . , . I p7f(x,) = l 1x. 1 xm2 ':' x0 xo2 ... Xn Xl Xl ... ymrn-l 2 m f(XfJ m ,.. * . . 1 I . . x,’ xm X”, m NOW we shall deduce an expression forf(x,) by aid of the divided differences.
The roots rr , r2, . . , r, of the polynomial q(f) are all real, but there are multiple roots. ; where m,. may be equal td 1, 2, 3, * * . , and so on. The general formula of decomposition into partial fractions is the following: If we multiply this expression by (f-ry)m~ we get (4) 233 (f---ry)Nlt, = t&l (f--fTy%-l + . . a,em,, + R(f) , (f-r,)“‘v WI We have l)(f) = C(f-rp (f--r2)m~*, . (t), A,(r,) will be different from zero: A”(f”) = am, In the same manner the first m,,--s derivatives give D”~-~dv(rJ = (m,-s)!