Calculus of Finite Differences (AMS Chelsea Publishing) by Charles Jordan

By Charles Jordan

Booklet by means of Charles Jordan

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F,,. (DY,,) ‘I (F )“” ’ . , , RnD”vo ( n! 1 - 0 13. Expansion of functions by aid of decomposition into partial fractions. /p(t). We may always suppose that ,, (t) and (,1(f) have no roots in common; since if they had, it would always be possible to simplify the fraction, dividing by f-r,,, , if r,,, i s . the root. A. Let us suppose that the roots t,, r3, , , , , r,, of t/*(f) are all real and unequal. We have u= tt+l ai z a=, f--c 35 Reducing the fractions to a common denominator we obtain u if for every value of t we have *+1 44 Y(t) = iz, Qi-ri therefore this is an identity: so that the coefficients of P’, in both members, must be equal, This gives R equations of the first degree which determine the coefficients oi.

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. First multiplying sf (x,) by w. (x,,J, then B2f (x0) by o1 (x,) and so on: jDvf (x,) by wIW1 (x,,) and finally Bmf (x,,) by CO,,,-, (x,) we obtain + m+1 $1 cd”-1 = i=l v=i D (x,,,) wy (Xi) f(xi)W . =~ D toy (x,) = -’ ’ so that the coefficient of f(x,,) is equal to -1. Moreover it can be demonstrated that VI+1 WY-l (&) ,zl D (tiv (Xi) = O if m ’ i ’ O therefore for these values the term f(xi) will vanish from the equation and we have f M = f (x0) + h---x,) w (4 + brr-x0) (x,,,--xl) 6’f(X”) + (4 + bi---x,1 hr--XI) (w---x,) %3f(X,,) + ’ * * ’ + hr-4 I&c--xl) * * - (xm--x,-l) 9”f(X”).

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. 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 ,..

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