{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 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 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 "" 0 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 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 }{PSTYLE "" 0 259 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 258 "" 0 "" {TEXT -1 23 "Die Methode von Hermite " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "resta rt; " }}}{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 "" {TEXT -1 92 "Abspalten des polynomialen Anteils : f wird ersetzt durch den Divisionsrest von Z\344hler/Nenner" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := sort(rem(f, g, x, 'quot'));" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Quotient:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "quot;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Damit ha ben wir einen ersten Teil des Integrals:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Integral := int(quot, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Quadratfreie Faktorisierung des Nenners:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(g, sqrfree);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "Da beide Faktoren Grad vier haben und der polynomiale An teil des Integranden bereits abgespalten ist, k\366nnen wir einen Ansa tz mit Z\344hlern vom Grad drei machen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h1 := alpha[3]*x^3 + alpha[2]*x^2 + alpha[1]*x + alpha[0];" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "h2 := beta[3]*x^3 + beta[2] *x^2 + beta[1]*x + beta[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Ansatz := (f/g - h1/(x^4-1) - h2/(x^2-2)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Zaehler := collect(sort(expand(numer(normal(A nsatz)))), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "L\366sung des l inearen Gleichungssystems, das alle Koeffizienten auf Null setzt:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(\{coeffs(Zaehler, x)\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Die Z\344hler sind also" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "h1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "h2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "normal(f/g - h1/(x^4-1) - h2/(x^2-2)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Der Hermitesche Algorithmus mu\337 nur au f den zweiten Summanden angewandt werden, also auf" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "h2/(x^2-2)^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Wir kombinieren den ggT des Nenners mit seiner Ableitung linear:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gcdex(x^2-2, 2*x, 'x', 'a', 'b'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a; b;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a*( x^2-2) + b*(2*x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Der Z\344hle r des im n\344chsten Schritt noch zu integrierenden Bruchs mit Nenner \+ x^2 - 2 ist" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rest := a*h2 + diff( b*h2, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Die Funktion, die wi r bereits abspalten und zur Stammfunktion addieren k\366nnen, ist" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sort(expand(b*h2))/(x^2-2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Integral := expand(Integral \+ - %);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "normal(diff(Integral, x) + h1/(x^4-1) + Rest/(x^2- 2));" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 32 "Die Methode von Rothste in-Trager" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "F \374r Rothstein-Trager gibt es also zwei Summanden. Beginnen wir mit" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "h1/(x^4-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "resultant(h1 - c*4*x^3, x^4-1, x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcd(h1 + x^3, x^4-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcd(h1 - x^3, x^4-1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "gcd(h1 - ((I/9)*x^3), x^4-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gcd(h1 + ((I/9)*x^3), x^4-1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Damit ist dieser Term erledigt, u nd wir k\366nnen die vier gerade gefundenen logarithmischen Terme zur \+ Stammfunktion addieren:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Integral := Integral - (1/4)*log(x+1) + (1/4)*log(x-1) + (I/36)*log(x+I) - (I/ 36)*log(x-I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Bleibt noch der \+ letzte Term" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Rest/(x^2-2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Hier mu\337 zun\344chst ein polyno mialer Anteil abgespaltet werden:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rest := rem(Rest, x^2-2, x, 'quot');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "quot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "In tegral := Integral + int(quot, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Jetzt kann Rothstein-Trager angewandt werden:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "resultant(Rest - c*2*x, x^2-2, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gcd(Rest - 22/9*sqrt(2) * 2*x, x^2-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gcd(Rest + 22/9*sqrt(2) * 2*x, x^2- 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Integral := Integral + 22/9*sqrt(2)*log(x - sqrt(2)) - 22/9*sqrt(2)*log(x + sqrt(2));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(Integral, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "De r Z\344hler stimmt. Den Nenner m\374ssen wir erst noch ausmultiplizier en:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sort(expand(denom(%)));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Vergleich mit dem Original:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "g;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Nat\374rlich l\344\337t sich das Integral auch sch\366ner ausdr \374cken:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(x^10/g, x);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Auch das wird noch \374bersichlich er, wenn man etwas umordnet und die beiden Logarithmen zusammenfa\337t :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "1/3*x^3+4*x-8/3*x/(x^2-2)+1/4*ln((x-1)/(x+1)) +1/18*arctan(x)-44/9 *sqrt(2)*arctanh(1/2*sqrt(2)*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(arctan(x), ln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(arctanh(x), ln);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "1/2 * ln((x+1)/(x-1));" }}}{EXCHG {PARA 256 "" 0 " " {TEXT 256 17 "Zweites Beispiel:" }}{PARA 257 "> " 0 "" {MPLTEXT 1 0 15 "h := x^5+2*x+1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Wir suchen " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Int(1/h, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Der Nenner ist quadratfrei:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "gcd(h, diff(h, x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Wir k\366nnen also gleich mit Rothstein-Trager beginnen: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "c := 'c': resultant(1 - 'c'*dif f(h, x), h, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "c := sol ve(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Diese L\366sung ist al so nicht durch Wurzeln ausdr\374ckbar, und wir m\374ssen im Zerf\344ll ungsk\366rper dieses Polynoms rechnen. Dazu empfiehlt sich zun\344chst eine Abk\374rzung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "alias(zeta=R ootOf(11317*_Z^5-1280*_Z^3-320*_Z^2-30*_Z-1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Im K\366rper " }{TEXT 257 1 "Q" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "zeta;" "6#%%zetaG" }{TEXT -1 69 ") k\366nnen wir rechne n: Wir fassen ihn auf als einen f\374nfdimensionalen " }{TEXT 258 1 "Q " }{TEXT -1 22 "-Vektorraum mit Basis " }{XPPEDIT 18 0 "\{1, zeta, zet a^2, zeta^3, zeta^4\};" "6#<'\"\"\"%%zetaG*$F%\"\"#*$F%\"\"$*$F%\"\"% " }{TEXT -1 65 " und beachten bei der Multiplikation, da\337 sich alle Potenzen von " }{XPPEDIT 18 0 "zeta;" "6#%%zetaG" }{TEXT -1 130 " ab \+ der f\374nften durch die Basiselemente ausdr\374cken lassen. Insbesond ere funktioniert der Euklidische Algorithmus und wir erhalten " } {MPLTEXT 1 0 30 "\n gcd(1 - zeta*diff(h, x), h);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 183 "Wir m\374ssen also \374ber die f\374nf Wurzeln di eses irreduziblen Polynoms summieren und erhalten jeweils einen Summan den der Form Wurzel mal Logarithmus des gerade berechneten Linearfakto rs." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Sum(zeta *ln(x+4096/625+93237/625*zeta+564824/625*zeta^2+724288/625*zeta^3-5794 304/625*zeta^4), zeta);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Im wesentlichen genauso arbeitet der eingebaute Integrat or von Maple. Wir machen zun\344chst die Abk\374rzung " }{XPPEDIT 18 0 "zeta;" "6#%%zetaG" }{TEXT -1 13 " r\374ckg\344ngig: " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "alias(zeta=zeta);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "int(1/h, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }