{VERSION 5 0 "HP RISC UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 24 "Die Methode von Horowitz " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Zun\344chst geht alles gen auso, wie bei Hermite:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "f := x^10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g := x^8+3*x^4-4*x^6+4*x^2-4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Integrand := f/g;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "P := quo(f, g, x, 'Rest');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "polyAnteil := int(P, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f := Rest;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Jetzt berechnen wir die Nenner g1 des rationalen und g2 des logari thmischen Anteils" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g1 := gcd(g, diff(g, x));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "g2 := quo(g, g1, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "F\374r die Z\344hler, deren Grad kleiner ist, machen wir \+ einen Ansatz mit unbestimmten Koeffizienten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c := 'c': f1 := c[0] + c[1]*x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "d := 'd': f2 := sum(d[i]*x^i, i=0..5);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Die rechte Seite der Horowitzsch en Gleichung ist" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "RechteSeite := \nsimplify(diff(f1, x)*g2 - f1*diff(g1 , x)*g2/g1 + f2*g1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Dies soll gleich f sein; wir berechnen die Differenz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "collect(sort(simplify(f-RechteSeite), x), x);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Alle diese Koeffizienten m\374ss en verschwinden:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "LGS := \{coeffs(%, x)\}:\nfor i to nops(LGS) do LGS[i ]=0 od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(LGS);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Die beiden Z\344hler sind also" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "f1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(f/g - diff(f1/g1, x) - f2/g2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "RatAnteil := simplify(f1/g1);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "logAnteil := simplify(f2/g2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Darauf wenden wir Rothstein-Trager an:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := so rt(numer(logAnteil));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g \+ := sort(denom(logAnteil));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "c := 'c': Res := resultant(f - 'c'*diff(g, x), g, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(Res);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "c := [solve(Res)];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "for k to nops(c) do\nv[k] := gcd(f - c[k]*diff(g, x ), g) od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sum(c['k']*log (v['k']), 'k'=1..nops(c));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Add ition aller Beitr\344ge:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Integral := polyAnteil + RatAnteil + sum(c['k']*lo g(v['k']), 'k'=1..nops(c));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Pro be:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "s implify(diff(Integral, x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "normal(numer(%)/expand(denom(%)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }