NOT E ON DIVIDED DIFFERENCE S

Transcription

1 Det Kgl. Danske Videnskabenes Selskab. Mathematisk-fysiske Meddelelse. XVII, 3. NOT E ON DIVIDED DIFFERENCE S SY J. F. STEFFENSEN KØBENHAVN EJNAR MUNKSGAAR D 939

2 Pinted in Denmak. Bianco Lunos Bogtykkei A/S.

3 . The numbe of geneal theoems concening divide d diffeences is so small that any addition to the list may, pehaps, be welcome. The unexpectedly simple theoe m which foms the object of this Note seems, as fa as I have been able to ascetain, to be new ; it may be egaded a s a genealization of LEIBNIZ' fomula fo the t}' deivate o f a poduct of two functions. The notation will be that of the autho's book "Intepolation". Thus, fo instance, (xo x... x,) will be the '' divided diffeence of p (x), fomed with the aguments xo, x,... x. In ode to save space we shall, as a- ule, only wite the fist and the last of the aguments, whee n o confusion is likely to aise. Let, then, (x) = f(x) g (x) ; ( ) we popose to pove, by induction, that F (xo... x ) f(xo... x ) g (xv... x,.) ( 2) It is eadily ascetained that the fomula is tue fo =, that is, p (x o x ) = f(xo) 9(x o xi) + f(xo x) g (x) We poceed to show that, if the fomula is tue fo on e value of, it also holds fo the following value. *

4 4 N. 3. J. F. STEFFENSEN : In ode to pove this, we employ the identity y (x0 -.. x) - (x... x +I) (x 0... x+) = x0-x+ Applying this, we find, assuming (2) to be tue fo som e paticula value of, `x0-x+) (x0.. x+ ) _ l (x0... x) g (xv... x) f(x... xv + ) g (xv +... x + ). =o - o Inseting, in this, { f(x... x y + = f(x0... x) -(x0- xy-i -)! (x 0... xy +), we find. (x0 -x +i) y (x0... x +) f. x) g (xy... x) - f (x0... x) g (xv +... x + v= 0 - x y+) f(x0... xy+i)g(xy+l... x +). In the second sum on the ight we intoduc e g(x9+...x+) = g(xy...x) -(xv-x+g(xi,...x--), and in the thid sum we wite v- instead of v. Thus, we obtain x0 -x +l) ~(x0... x +) _ f... x ) g (xy... x) - ~ f(xo... x) g (x,,... x ) `x z -x+)f(xo... x) g(xv... x+l) - x,.,) f(x o... x ) g (xv... x + )

5 Note on Divided Diffeences. 5 which educes to + so that (xo - xt, +) f(xo... xv) g(xi,... xl, + i), _ o + / ~ (xo... x+) - ~ fl xo... x,,) g(x... x+. But this is (2) with + instead of, so that (2) is tue fo all values of. 2. Fomula (2) contains seveal well-known fomula s as paticula cases. Thus, if we make all the aguments xz, tend to the same point x, we obtain, if the deivates exist, 90(0 (X) ' f(v ) (x) g t vl (-v) v= o which may also be witte n (-v) (x) D f (x) g (x) _ ` v ) Dv f(x). D-v g (x) ( 3) This is the theoem of LEIBNIZ efeed to above. Putting next, in succession, xv = x -}- v, x, = x - v and x,, = x- 2 + v and making use of the elation s f(x, x+,.., x+ n) f(x, x-,..., x-n) L n f(x) 2 t v n f( x) l2 : - 2 -F,..., x + 2 å n h(x) we obtain, in analogy with (3), the thee well-known elations

6 6 N. 3. J. F. STEFFENSEN : A f(x) g(x) _() f(x) ~ y g (x + v), (4) v= o O f(x) g (x) _,), f (Iv') o' f(x} o ' g(xv), v (5) (x- 2 ") å-vg ( x+ v ) (6) 3. We now put so that and f(x) = F(t)-F(x), g( x) = x, g(xy... x) (t-xv)... (t-x,.) F(t)- F(x) t- x ( 7) ( 8) Inseting in (2), we obtain, keeping the fist tem on th e ight apat, F(t)-F(xo) \ ' F(xo... x v) (xo... x ) _ (t-xo)... (t-x ) ( t-x v)... (t-x) o, solving fo F(t), F(t) = t -xo)... (t-xv i )F(.xo... x v )+R, ( 9) R = (t-xo)... (t- x) P (xo... x), (0) whee the factoial (t-xo)... (t-xv_i) fo v = 0 is intepeted as. This is NEWTON' S intepolation fomula with divide d diffeences and a emainde tem diffeing slightly fo m the usual fom. The latte is obtained by obseving that, if we put

7 Note on Divided Diffeences. 7 ~p f(xo) = f(xo xp) ; Of(x o) = f(xo 0, () 8 p and 6 being symbols acting on xo alone, then, sinc e (xo) = Ø F (xo), (xo... x) = B O... Bt P (x o) = BB_ i... 9 B F (x o) o T(xo... x i.) = F(txo... x), (2) so that R = (t-xo)... (t-x.) F(txo... xe). (3) But fom (0) we obtain in paticula cases foms o f the emainde which ae woth noting. Thus, fo instance, if all the aguments tend to the same point x, we fin d TAYLOR ' S fomula with the emainde F(t) _ \ ' (f-x)v F(Y) (x) + R (4) v= 0 R _ (t-x)-- D F(t) - F(x)! t- x (5 ) the opeato D acting on x. Futhe, putting xi, = x+ v, (9) and (0) yiel d F(t) (t-x) (") G v F(x)-I-R, (6) R = (t-x) (+ ~ ) G F(t)-F(x )! t-x ' (7) whee G acts on x. This is the intepolation fomula wit h descending diffeences and a emainde tem which ha s aleady been given by BooLE'. Finite Diffeences, 3 d ed., p. 46.

8 8 N. 3. J. F. STEFFENSEN : Finally, putting x = x - v, we find the intepolation fomula with ascending diffeence s acting on x. F(t) _'(t- x)(-") v! R = (t-x) VyF(x)-I-R, (8) v F (t )-F (x)! t-x (9) It is evidently easy to tansfom the peceding emain - de tems to the usual foms. 4. It is easy to extend the fomula (2) to a poduct o f any numbe of functions. Thus, if we have f(x) = fy (x) f2 (x), v g (x) = f3 (x), f'(xo... x ) = fi (xo... xd fs (x u... x,,), = and (x) - f (x) f2 (x)f3(x) 9) (xo. x ) =- h. (xo. x i o) f2 (x~~... xv) f3 (xv... x ). P (x) = A. (x) f2 (x)... fn (x), ( 2 0) cp (xo... x) _ fl x,,) f2 (xa... xf?) fj (xß... xy)... fn (xp... x), (2 ) the summation extending to all values of a, 48, y,... Q fo which 0 <a<,g <y< <e<, (22)

9 Note on Divided Diffeences. 9 Thus, fo instance, if n = 3 we may at once wite down ~ (x0 x x2) = (x0).2 (x0) f3 (x0 xl x 2) + ft (x 0) f2 (x0 x) f3 (xt x2) f (x 0) f2 (x0 xl x2) f3 (x 2 ) - I+ + f (xo xl) f2 (xi) f3 (xi x2) + fl (x o xl) f2 (x x2) f3 (x 2) + fl (x0 x i x 2) f2 (x 2) f3 (x 2) If, in (2), we let all the aguments tend to the sam e point x, we ge t T () (x) ' f (a) P) (x) f2 (ß-a) (x) (x)... fn! a! (d--a)! (-~O) I and fom this, putting a = v, fi-a = v2,..., -~O = v n, cy () (x) =! v l! v2!... vn! fi(vl) (x) f2(vd(x)... fn("n)(x), (23) the summation extending to all values of vl, v2,.., vn fo which yl + +. y2.. + vn =. (24) This is the theoem of Leibniz fo a poduct of n functions. It may he witten symbolically in the fo m P(7) - (fl + f fn) (25) with the convention that, afte expanding, should b e eplaced by fw). It should be noted that the zeo powes of f cannot be omitted, since f(0) does not mean but f. If, in (2), we choose x,, = x + v, we find,cso(x) L afl(x),l ß-a f2(x + a) L -`ofn(x+ P)! cc! (,6-a)! ( -Q)! o, in the notation (24),

10 0 N. 3. J. F. STEFFENSEN : ~p(x) - v I Y 2 L... Yn l Î X (26) Q Yl fi (x) G v2 f2 (x + Similaly, putting x, = x - v, we obtain V p(x) =t I Yl Iv2!..v,t ~ 7 'f(x)v "2 f2 (x.-yl)... D"nfn(x-vl-... yn X (27) and finally, making x,, = x- 2 -f- v, ~// lv l'y!~ 2 fyl J. ~", f x +vl+...+ vn -I'- 7n ~ ). 2 It is easy to put also (26), (27) and (28) into symboli c foms ; but as these ae moe complicated than (25) and, theefoe, not so useful, they seem hadly woth ecoding. 5. As an application of (2) we put f (x) = p(x) = (29) t-x (t-x) and obtain without difficulty (xo...x) =. (t-x) (t-xa) (t-xp). (t-x v) the summation extending to the values of a,,ß,... e satisfying (22).

11 Note on Divided Diffeences. But since the degee of the poduc t (t-x,) (t-xß)... (t-x o) is the same as the numbe of the quantities a, ß,... which is n-, (30) may also he witten 9) (xo... x) _ i _ \JJ }(3) (t - x 0)... (t - xi.) (t x0) NCO... (t - x) No JI the summation extending to all values of tto,. which po+tt P, = n-. p, fo (32) whee Instead of (3) and (32) we may evidently wit e P(xo... x) = (33) (t-x0) Z0... (t-x) tio -I fi = n±, >. (34) It thus appeas that y (x 0.. x) is the coefficient o f zn -F in the development o f z z z t-x0 t xi t- z ) (- t \ tzxol ( - zx, ) (35) o the coefficient of z L in the development o f (t-x0 -z) (t-xl -z)... (t-x-z). (36) The numbe of tems in (33) is obtained by puttin g t =, x = 0 fo all v, and is theefoe, accoding to (36), the coefficient of zn in the development of (-z) -i., that is, (+n- n- ) '

12 2 N. 3. J. F. STEFFENSEN : Note on Divided Diffeences. 6. Lastly, we conside the cas e f (x) = x! (x) = (-4 x) (ttt -x), (37 ) assuming all tv diffeent. Hee, an abbeviation of the notation becomes necessay, and we shall wit e to = (t-x e ) (t-xn+i)... (t-xd. (38) We obtain, then, fom (2) (x0... x) = t0a t2... tn s (39 ) the summation extending as befoe to (22). But we have also, fo potation fomula, instance by LAGRANGE ' S inte - (x) = _ (tl -x)... (tn-x) (t -x) (40) whee K = (t om i ) (t2-ty) (t + -t )... (tn-ç), (4) so that P (x 0... x) _ (42) We ' theefoe obtain, by compaison of (42) and (39), th e identity ~ t u t a ~... tf~ L.! li t O 2 n y In the paticula case whee n = 2 > _ t Ovtv t -t t i to ) this become s (43) (44) Indleveet til Selskabet den 3. Mats 939. Fædig fa Tykkeiet den 22. Juni 939.