By Jordan C.
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Additional resources for Calculus of Finite Differences
X, may be anything whatever. 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.
Q 9. Divided differences. So far we have supposed that the function f(x) is given for x=x,, x,, x,, . , x,, and that the interval xi+1 - xi = h deal with the general problem is independent of i. Now we where the system of x,,, x,, . . , x, may be anything whatever. 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.
Formula (3) may be written f(x) = r(x,,) + !. o;(x) zi+‘f (x,,) + ,mwr;,,‘, Dm+‘fC1, Q 10. Generating functions. One of the most useful methods of the Calculus of Finite Differences and of the Calculus of 21 Probability is that of the Generating Functions, found by Laplace and first published in his “Theorie Analytique des Probabilites” [Courtier, Paris 1812). Given a function f(x) for x=a, a+l, a+2,. . , b-l u(t) will be defined as the generating function of f(x) if the coefficient of tX in the expansion of u(t) is equal to f(x) in the interval a, b.