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{SECT 0 {EXCHG {PARA 222 "" 0 "" {TEXT 250 37 "7.2A. Bogenl\344nge von
 Potenzfunktionen" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 254 135 "r
estart:\ninterface(warnlevel=0,showassumed=0):\nwith(plots): \nwith(pl
ottools): \nsetoptions(numpoints=300,scaling=constrained,axes=none):" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "In diesem Erg\344nzungsabschnit
t wollen wir uns der Frage widmen, f\374r welche rationalen Exponenten
 " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 22 " die Potenzfunktionen \+
" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "f(x) = x^
k;" "6#/-%\"fG6#%\"xG)F'%\"kG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 114 "eine elementar bestimmbare Bogenl\344nge besitzen. Eilige Lese
r k\366nnen diesen Abschnitt \374bergehen (oder \374berfliegen)." }}
{PARA 0 "" 0 "" {TEXT 256 18 "Die Ableitung von " }{XPPEDIT 18 0 "f(x)
;" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " " }{TEXT 256 14 "ist\n          " }
{XPPEDIT 18 0 "`f\264`(x) = k*x^`k-1`;" "6#/-%#f|_vG6#%\"xG*&%\"kG\"\"
\")F'%$k-1GF*" }{TEXT -1 62 " ,\nund das Integral f\374r die Bogenl
\344nge lautet daher\n          " }{XPPEDIT 18 0 "s[a](b) = int(sqrt(1
+k^`2`*x^`2k-2`),x = a .. b);" "6#/-&%\"sG6#%\"aG6#%\"bG-%$intG6$-%%sq
rtG6#,&\"\"\"F2*&)%\"kG%\"2GF2)%\"xG%%2k-2GF2F2/F8;F(F*" }{TEXT -1 1 "
 " }{TEXT 256 60 " .\nAuf den ersten Blick sieht das nicht gar so schl
imm aus. " }{TEXT -1 48 "Erstaunlicherweise ist die Aufgabe aber nur f
\374r " }{XPPEDIT 18 0 "k = 1;" "6#/%\"kG\"\"\"" }{TEXT -1 22 " sowie \+
f\374r\n           " }{XPPEDIT 18 0 "k = (n+1)/n;" "6#/%\"kG*&,&%\"nG
\"\"\"F(F(F(F'!\"\"" }{TEXT -1 7 "  und  " }{XPPEDIT 18 0 "k = n/(n+1)
;" "6#/%\"kG*&%\"nG\"\"\",&F&F'F'F'!\"\"" }{TEXT -1 26 "  \nmit nat
\374rlichen Zahlen " }{TEXT 271 1 "n" }{TEXT -1 60 " einigerma\337en l
eicht l\366sbar, w\344hrend alle anderen Werte von " }{TEXT 270 1 "k" 
}{TEXT -1 29 " vehemente Probleme bereiten." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 163 "print(x^k, k=1..20);\ndisplay(seq(display(rectang
le([0,0],[1,1]),plot(x^k,x=0..1,color=COLOR(RGB,1-1/k,0,1/k))),k=1..20
),insequence=true,\nview=[-.1..1.1,-.1..1.1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$)%\"xG%\"kG/F%;\"\"\"\"#?" }}{PARA 13 "" 1 "" 
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ie Neilsche Parabel heraus. Nachdem wir diese beiden F\344lle bereits \+
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\"Q!Fhf^n-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"\"Fcg^n$\"#6Fcg^nFag^n" 1 
2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve
 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" 
"Curve 16" "Curve 17" "Curve 18" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Hier erhalten
 wir die Ableitung \n           " }{XPPEDIT 18 0 "`f\264`(x) = k*x^`1/
n`;" "6#/-%#f|_vG6#%\"xG*&%\"kG\"\"\")F'%$1/nGF*" }{TEXT -1 46 " ,  \n
\ndie skalare Geschwindigkeit\n\n           " }{XPPEDIT 18 0 "v(x) = s
qrt(1+k^`2`*x^`2/n`);" "6#/-%\"vG6#%\"xG-%%sqrtG6#,&\"\"\"F,*&)%\"kG%
\"2GF,)F'%$2/nGF,F," }{TEXT -1 85 ",\n\nund das unbestimmte Integral z
ur Berechnung der Bogenl\344nge ist wieder \n           " }{XPPEDIT 
18 0 "int(v(x),x);" "6#-%$intG6$-%\"vG6#%\"xGF)" }{TEXT -1 27 " .\nBet
rachten wir den Fall " }{TEXT 268 7 "gerader" }{TEXT -1 8 " Zahlen " }
{XPPEDIT 18 0 "n = 2*m;" "6#/%\"nG*&\"\"#\"\"\"%\"mGF'" }{TEXT -1 47 "
. \nWir substituieren die Wurzel und l\366sen nach " }{TEXT 264 1 "x" 
}{TEXT -1 18 " auf:\n\n           " }{XPPEDIT 18 0 "v = sqrt(1+k^`2`*x
^`1/m`);" "6#/%\"vG-%%sqrtG6#,&\"\"\"F)*&)%\"kG%\"2GF))%\"xG%$1/mGF)F)
" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "x = (v^2-1)^m/(k^`2m`);" "6#/%\"x
G*&),&*$%\"vG\"\"#\"\"\"F+!\"\"%\"mGF+)%\"kG%#2mGF," }{TEXT -1 4 " ,  \+
" }{XPPEDIT 18 0 "dx = m*(v^2-1)^(m-1)/(k^`2m`)*2*v*dv;" "6#/%#dxG*.%
\"mG\"\"\"),&*$%\"vG\"\"#F'F'!\"\",&F&F'F'F-F')%\"kG%#2mGF-F,F'F+F'%#d
vGF'" }{TEXT -1 50 " .\nDamit formen wir das Integral um zu\n         \+
  " }{XPPEDIT 18 0 "2*m/(k^(2*m));" "6#*(\"\"#\"\"\"%\"mGF%)%\"kG*&F$F
%F&F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int((v^2-1)^(m-1)*v^2,v);
" "6#-%$intG6$*&),&*$%\"vG\"\"#\"\"\"F,!\"\",&%\"mGF,F,F-F,*$F*F+F,F*
" }{TEXT -1 55 " .\nUnter Verwendung der binomischen Formel \n        \+
   " }{XPPEDIT 18 0 "(z-1)^(m-1) = sum(B(m-1,j)*z^j*(-1)^(m-1-j),j = 0
 .. m-1);" "6#/),&%\"zG\"\"\"F'!\"\",&%\"mGF'F'F(-%$sumG6$*(-%\"BG6$,&
F*F'F'F(%\"jGF')F&F3F'),$F'F(,(F*F'F'F(F3F(F'/F3;\"\"!,&F*F'F'F(" }
{TEXT -1 46 "\nmit den Binomialkoeffizienten    \n           " }
{XPPEDIT 18 0 "B(m-1,j) = (m-1)!/j!/(m-j-1)!;" "6#/-%\"BG6$,&%\"mG\"\"
\"F)!\"\"%\"jG*(-%*factorialG6#,&F(F)F)F*F)-F.6#F+F*-F.6#,(F(F)F+F*F)F
*F*" }{TEXT -1 59 " \ngelangen wir durch gliedweise Integration zu\n  \+
          " }{XPPEDIT 18 0 "2*m/(k^(2*m));" "6#*(\"\"#\"\"\"%\"mGF%)%
\"kG*&F$F%F&F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sum(B(m-1,j)*(-1)
^(m-1-j)*int(v^(2*j+2),v),j = 0 .. m-1);" "6#-%$sumG6$*(-%\"BG6$,&%\"m
G\"\"\"F,!\"\"%\"jGF,),$F,F-,(F+F,F,F-F.F-F,-%$intG6$)%\"vG,&*&\"\"#F,
F.F,F,F9F,F6F,/F.;\"\"!,&F+F,F,F-" }{TEXT -1 15 " =\n            " }
{XPPEDIT 18 0 "(2*m)^(2*m+1)/((2*m+1)^(2*m));" "6#*&)*&\"\"#\"\"\"%\"m
GF',&*&F&F'F(F'F'F'F'F'),&*&F&F'F(F'F'F'F'*&F&F'F(F'!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "sum(B(m-1,j-1)*(-1)^(m-j)*v^(2*j+1)/(2*j+1),j = \+
1 .. m);" "6#-%$sumG6$**-%\"BG6$,&%\"mG\"\"\"F,!\"\",&%\"jGF,F,F-F,),$
F,F-,&F+F,F/F-F,)%\"vG,&*&\"\"#F,F/F,F,F,F,F,,&*&F7F,F/F,F,F,F,F-/F/;F
,F+" }{TEXT -1 44 " .        \nF\374r die Bogenl\344nge der Funktion  \+
" }{XPPEDIT 18 0 "x^k;" "6#)%\"xG%\"kG" }{TEXT -1 7 "  mit  " }
{XPPEDIT 18 0 "k = (2*m+1)/(2*m);" "6#/%\"kG*&,&*&\"\"#\"\"\"%\"mGF)F)
F)F)F)*&F(F)F*F)!\"\"" }{TEXT -1 77 "  zwischen 0 und 1 erhalten wir d
aher nach der R\374cksubstitution\n\n            " }{XPPEDIT 18 0 "v(x
) = sqrt(1+(2*m+1)^2/((2*m)^2)*x^`1/m`);" "6#/-%\"vG6#%\"xG-%%sqrtG6#,
&\"\"\"F,*(,&*&\"\"#F,%\"mGF,F,F,F,F0*$*&F0F,F1F,F0!\"\")F'%$1/mGF,F,
" }}{PARA 0 "" 0 "" {TEXT -1 55 "\ndas folgende bestimmte Integral:   \+
       \n           " }{XPPEDIT 18 0 "s[2*m];" "6#&%\"sG6#*&\"\"#\"\"
\"%\"mGF(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(v(x),x = 0 .. 1);" "6
#-%$intG6$-%\"vG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 23 " \n             \+
     =  " }{XPPEDIT 18 0 "(2*m)^(2*m+1)/((2*m+1)^(2*m))" "6#*&)*&\"\"#
\"\"\"%\"mGF',&*&F&F'F(F'F'F'F'F'),&*&F&F'F(F'F'F'F'*&F&F'F(F'!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "sum(B(m-1,j-1)*(-1)^(m-j)/(2*j+1),j = 1
 .. m);" "6#-%$sumG6$*(-%\"BG6$,&%\"mG\"\"\"F,!\"\",&%\"jGF,F,F-F,),$F
,F-,&F+F,F/F-F,,&*&\"\"#F,F/F,F,F,F,F-/F/;F,F+" }{TEXT -1 3 " ( " }
{XPPEDIT 18 0 "sqrt(1+(2*m+1)^2/((2*m)^2))^(2*j+1)-1;" "6#,&)-%%sqrtG6
#,&\"\"\"F)*&,&*&\"\"#F)%\"mGF)F)F)F)F-*$*&F-F)F.F)F-!\"\"F),&*&F-F)%
\"jGF)F)F)F)F)F)F1" }{TEXT -1 22 ")\n                  = " }{XPPEDIT 
18 0 "sum(B(m-1,j-1)*(-1)^(m-j)/(2*j+1)*(2*m)^(2*m-2*j)/((2*m+1)^(2*m)
),j = 1 .. m);" "6#-%$sumG6$*,-%\"BG6$,&%\"mG\"\"\"F,!\"\",&%\"jGF,F,F
-F,),$F,F-,&F+F,F/F-F,,&*&\"\"#F,F/F,F,F,F,F-)*&F5F,F+F,,&*&F5F,F+F,F,
*&F5F,F/F,F-F,),&*&F5F,F+F,F,F,F,*&F5F,F+F,F-/F/;F,F+" }{TEXT -1 2 " (
" }{XPPEDIT 18 0 "sqrt(z)^(2*j+1)-(2*m)^(2*j+1);" "6#,&)-%%sqrtG6#%\"z
G,&*&\"\"#\"\"\"%\"jGF,F,F,F,F,)*&F+F,%\"mGF,,&*&F+F,F-F,F,F,F,!\"\"" 
}{TEXT -1 30 ") ,\n                          " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "wobei  " }{XPPEDIT 18 0 "z = (2*m)^2+(2*m+1)^2;" "6#/%\"zG
,&*$*&\"\"#\"\"\"%\"mGF)F(F)*$,&*&F(F)F*F)F)F)F)F(F)" }{TEXT -1 48 " g
esetzt wurde. Auf jeden Fall ist\n             " }{XPPEDIT 18 0 "s[2*m
];" "6#&%\"sG6#*&\"\"#\"\"\"%\"mGF(" }{TEXT -1 4 " =  " }{XPPEDIT 18 
0 "q[m]*sqrt(z)+r[m];" "6#,&*&&%\"qG6#%\"mG\"\"\"-%%sqrtG6#%\"zGF)F)&%
\"rG6#F(F)" }{TEXT -1 36 "   mit geeigneten rationalen Zahlen " }
{XPPEDIT 18 0 "q[m];" "6#&%\"qG6#%\"mG" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "r[m];" "6#&%\"rG6#%\"mG" }{TEXT -1 50 ".\n\nWir nehmen MAPLE zu Hil
fe, um diese Zahlen f\374r " }{TEXT 266 1 "m" }{TEXT -1 27 " = 1, ...,
 10 zu bestimmen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "for m
 from 1 to 10 do\ns[2*m]:=sum((m-1)!/(j-1)!/(m-j)!*(-1)^(m-j)/(2*j+1)*
(2*m)^(2*m-2*j)/((2*m+1)^(2*m))*(sqrt((2*m)^2+(2*m+1)^2)^(2*j+1)-(2*m)
^(2*j+1)),j = 1 .. m); od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"s
G6#\"\"#,&*(\"#8\"\"\"\"#F!\"\"F*#F+F'F+#\"\")F,F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%\"sG6#\"\"%,&*(\"%j<\"\"\"\"%v$*!\"\"\"#T#F+\"\"#F
+#\"%[?F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"',&*(\"'&)
G9\"\"\"\"'VN#)!\"\"\"#&)#F+\"\"#F+#\"''\\Y(\"(:x6%F-" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"),&*(\"*:uP$G\"\"\"\"+BM%>r#!\"\"\"$X
\"#F+\"\"#F+#\"+[O[Z@\",:rrfN\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%\"sG6#\"#5,&*(\".jlodS)=\"\"\"\"/$\\[_juz\"!\"\"\"$@##F+\"\"#F+#\".+
++++c#F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"#7,&*(\"1`#=Z
]!o$p\"\"\"\"\"2\"[.w?$Q@L#!\"\"\"$8$#F+\"\"#F+#\"1[!QjGiL/$F,F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"#9,&*(\"5P>'*z?66Z19\"\"\"
\"6v=n)Qt*yk&y=!\"\"\"$@%#F+\"\"#F+#\"5on\"Guy**QdF#F,F-" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"#;,&*(\"8b#HQ)>U7^;Y$f\"\"\"\":*z&\\:
arZq@eY1\"!\"\"\"$X&#F+\"\"#F+#\"9)3`te9t!)4HY/'\":&**yu2x&Q_3\"HB`F+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"#=,&*(\"<&zjHfPELE3Yz'z
#\"\"\"\"=\\HRg^LA-\"H=Q&4[!\"\"\"$&o#F+\"\"#F+#\"=ks)Ha')QjA'*>1'yD\"
>XZ'>!en66b94pZS#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"#?#
\"8F!He8Xr1hc\\O\"8B$**\\!G9;\\n)zD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 81 "\nDie letzte (rationale!) Zahl ist die Bogenl\344nge der beiden
 Kurven \n             " }{XPPEDIT 18 0 "x^`21/20`;" "6#)%\"xG%&21/20G
" }{TEXT -1 10 "  und     " }{XPPEDIT 18 0 "x^`20/21`;" "6#)%\"xG%&20/
21G" }{TEXT -1 83 "\nzwischen 0 und 1. Es passiert sehr selten, da\337
 die Bogenl\344nge von Kurven der Form " }{XPPEDIT 18 0 "x^k;" "6#)%\"
xG%\"kG" }{TEXT -1 109 " zwischen 0 und 1 eine rationale Zahl ist - ab
er es kommt unendlich oft vor, n\344mlich dann, wenn\n             " }
{XPPEDIT 18 0 "z = (2*m)^2+(2*m+1)^2;" "6#/%\"zG,&*$*&\"\"#\"\"\"%\"mG
F)F(F)*$,&*&F(F)F*F)F)F)F)F(F)" }{TEXT -1 69 " \nwieder eine Quadratza
hl ergibt. In diesem Fall ist die Bogenl\344nge  " }{XPPEDIT 18 0 "s[2
*m];" "6#&%\"sG6#*&\"\"#\"\"\"%\"mGF(" }{TEXT -1 4 " =  " }{XPPEDIT 
18 0 "q[m]*sqrt(z)+r[m];" "6#,&*&&%\"qG6#%\"mG\"\"\"-%%sqrtG6#%\"zGF)F
)&%\"rG6#F(F)" }{TEXT -1 11 " der Kurve " }{XPPEDIT 18 0 "x^((2*m+1)/(
2*m));" "6#)%\"xG*&,&*&\"\"#\"\"\"%\"mGF)F)F)F)F)*&F(F)F*F)!\"\"" }
{TEXT -1 28 " zwischen 0 und 1 rational. " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 99 "Da die Kurven sich eng an die Diagonale anschmiegen, soll
ten diese rationalen Bogenl\344ngen nahe bei " }{XPPEDIT 18 0 "sqrt(2)
;" "6#-%%sqrtG6#\"\"#" }{TEXT -1 53 " liegen. In der Tat ergibt numeri
sche Auswertung f\374r " }{XPPEDIT 18 0 "m = 10;" "6#/%\"mG\"#5" }
{TEXT -1 64 " immerhin eine \334bereinstimmung der ersten drei Nachkom
mastellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "S:=m->sum((m
-1)!/(j-1)!/(m-j)!*(-1)^(m-j)/(2*j+1)*(2*m)^(2*m-2*j)/((2*m+1)^(2*m))*
(sqrt((2*m)^2+(2*m+1)^2)^(2*j+1)-(2*m)^(2*j+1)),j = 1 .. m):\ns:='s':
\ns[20]=evalf(S(10));\nsqrt(2)=evalf(sqrt(2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"sG6#\"#?$\"+iJj99!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*$\"\"##\"\"\"F%$\"+iN@99!\"*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 281 "display(line([0,0],[1,1]),rectangle([0,0],[1,1]
),\nseq(plot(x^(k/(k+1)),x=0..1,color=COLOR(RGB,1-1/k,0,1/k)),k=20..20
),            seq(plot(x^((k+1)/k),x=0..1,color=COLOR(RGB,1-1/k,0,1/k)
),k=20..20),\nTEXT([-.04,-.02],'`(0,0)`'),TEXT([1.04,1.03],'`(1,1)`'),
view=[-.1..1.1,-.1..1.1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 130 130 
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G-%%VIEWG6$;$!\"\"Fhdt$\"#6FhdtFfdt" 1 2 0 1 10 0 2 9 1 1 1 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "          .\n\nE
in " }{TEXT 272 22 "pythagor\344isches Tripel" }{TEXT -1 30 " ist ein \+
Tripel ganzer Zahlen " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 3 " < \+
" }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "z;" 
"6#%\"zG" }{TEXT -1 142 ", die als Seitenl\344ngen eines rechtwinklige
n Dreiecks auftreten.\nDas ist nach dem Satz von Pythagoras genau dann
 der Fall, wenn \n              " }{XPPEDIT 18 0 "x^2+y^2 = z^2;" "6#/
,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'" }{TEXT -1 211 "\ngilt. Der p
olnische Mathematiker Waclaw Sierpinski hat in seinem Buch \"Pythagore
an Triangles\" 1962 alle pythagor\344ischen Tripel, bei denen sich die
 zwei kleineren Zahlen um 1 unterscheiden, bestimmt. Sind diese " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " und " }{XPPEDIT 18 0 "y = x
+1;" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 24 ", so mu\337 \n         \+
     " }{XPPEDIT 18 0 "x^2+(x+1)^2 = z^2;" "6#/,&*$%\"xG\"\"#\"\"\"*$,
&F&F(F(F(F'F(*$%\"zGF'" }}{PARA 0 "" 0 "" {TEXT -1 24 "gelten, und f
\374r gerades " }{XPPEDIT 18 0 "x = 2*m;" "6#/%\"xG*&\"\"#\"\"\"%\"mGF
'" }{TEXT -1 84 " ist das genau die Bedingung, die wir oben brauchten.
 Man beginnt mit \n             " }{XPPEDIT 18 0 "3,4,5;" "6%\"\"$\"\"
%\"\"&" }}{PARA 0 "" 0 "" {TEXT -1 45 "und bildet aus jedem schon gewo
nnenen Tripel " }{XPPEDIT 18 0 "x,x+1,z;" "6%%\"xG,&F#\"\"\"F%F%%\"zG
" }{TEXT -1 34 " ein neues, n\344mlich \n             " }{XPPEDIT 18 
0 "3*x+2*z+1,3*x+2*z+2,4*x+3*z+2;" "6%,(*&\"\"$\"\"\"%\"xGF&F&*&\"\"#F
&%\"zGF&F&F&F&,(*&F%F&F'F&F&*&F)F&F*F&F&F)F&,(*&\"\"%F&F'F&F&*&F%F&F*F
&F&F)F&" }{TEXT -1 6 ".\nAus " }{XPPEDIT 18 0 "3,4,5;" "6%\"\"$\"\"%\"
\"&" }{TEXT -1 37 " wird also zum Beispiel\n             " }{XPPEDIT 
18 0 "20,21,29;" "6%\"#?\"#@\"#H" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 21 "daraus \n             " }{XPPEDIT 18 0 "119,120,169;" "6%
\"$>\"\"$?\"\"$p\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 24 "und
 daraus\n             " }{XPPEDIT 18 0 "696,697,985;" "6%\"$'p\"$(p\"$
&)*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "Auf diese Weise en
tstehen " }{TEXT 273 4 "alle" }{TEXT -1 74 " solchen pythagor\344ische
n Tripel - ein genialer und schwieriger Satz!\nNach " }{XPPEDIT 18 0 "
21/20;" "6#*&\"#@\"\"\"\"#?!\"\"" }{TEXT -1 10 " ist also " }{XPPEDIT 
18 0 "697/696;" "6#*&\"$(p\"\"\"\"$'p!\"\"" }{TEXT -1 31 " der n\344ch
ste Exponent der Form " }{XPPEDIT 18 0 "k = (2*m+1)/(2*m);" "6#/%\"kG*
&,&*&\"\"#\"\"\"%\"mGF)F)F)F)F)*&F(F)F*F)!\"\"" }{TEXT -1 12 " , f\374
r den  " }{XPPEDIT 18 0 "x^k;" "6#)%\"xG%\"kG" }{TEXT -1 48 " eine rat
ionale Bogenl\344nge zwischen 0 und 1 hat." }}{PARA 0 "" 0 "" {TEXT 
-1 91 "\nTats\344chlich findet MAPLE durch exakte Auswertung der obige
n Summenformel f\374r das Integral " }{XPPEDIT 18 0 "s[696];" "6#&%\"s
G6#\"$'p" }{TEXT -1 39 " den \374ber eine Seite verteilten Bruch  " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "'s[696]'=S(348);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/
&%\"sG6#\"$'p#\"\\^pF%eWm2-EmcER%>snA:&*3NlCh$)4N!)fmdlgQz\\U6oN!=Ynpa
\\B;9,+(>50+3$*[AUNO?RUd$o@uSw_QrV>'G:dHmAGpS\"y.&[IX@dPdhSj/&)f\"*eOB
uG@Z9>T$\\@$GJ8O$*ol0P.0Oi*)y\"[ee_JV?JZ)*=+$fIW?1lB1)*))*Q<\"yBq;D6nJ
4N-k[sjlJ_X$Qh8^d#o!)4oPd%y1BJ[YA.2c&Hh_LO4&o8U^j%Q!H]Jj5'3oR^K'GZ&=_h
(R\"p>#z>zyi!R(o?.jKC2X@Couj+Jy8Fu<6Gq-\\]*e)>J)[22vFJ]'\\'fq*o\"y*\\j
:t,a1(oWqztM(\\];!3Xuu^o\\8eCoOUv3$4MiZ'\\RR+64]t!eV1aGEm7N9n%p*[mSoO2
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>zx>m$H6=\"eV;v@/#>CZm&\\$)Rl@))*epf\"G?-k)G`!\\Y:Z(z5-2T)*\\Qw;^c(o>>
d>6z?Zl5^Jx#\\H3kI%G#yr@OEnwiM*e$*Q*HW**3>4:!3&pQk/Er`UW!yD'y*Hr'o#QH3
mpxZu&3fJ$)\\;IcB!317^_E'H#GlSQ6zG3LQB<jP9u7H*Rg1?01^RQ_^sR#)HHL'yMhV6
+\"eO5]0_Fx\"fv1a3++cICoryFC*y7'HVT]bB,i)))\\?/8`-/b@Be1COZ02'eXM\\$yb
*pdk3&=t\\F(=\"\"[^pvo_?$)3flw,>#*>*otOD(o(=eS,))*Rh^.Np2<00qis)3#y3$G
W@j!\\T2k&R#RtMiOqQ\\:l_+k6Bc4(z%4v%*Q<L5wh!HZ@!zybg6Wo*4$fbP@'RaW4^#)
)eBS_vp8I6.YY2,%)oD2QUHjXt]>v^2H:FK)f1f'4[&)*\\pXYe)3hW$yF%*[3y<kKCVt#
GyM.*ecB'Hi0/1t4=`-IoQ_r-L(H3M(Q73%y\"y)=\"G51Qwf5Wfh2A)*G[\"3s^,(ebk[
4Af#4k)>b'*)4!Raf)o)GQ*=$)e]A>\\]$e/h$*)H8@DO0AXTNp9P;100#Qb3Md;*Q$z8I
!y+)4pvu$p!4-e*H7X,y.9#Q.&Hh\\OtjRa4mRi6b`22([(3l%**Rsl=IfqA?&QL58t2Ll
gH'\\GeI`=`8M?EJh&>74#33UeXJwd\\mO*zfk#4e5\\1;m4H3,e%G<T5>y^u:X?r\"Qk1
m=Q67*=*GV!H2g))ob7-y$)p)3\\F\\dki`8VL'o+hzb-*yp]**Gb()pj3?<O.EtN$3%>Q
0\\n\"\\93(fU$*ojM+Q#Q0r)QfU>*>B(fp25ybrLKo2.[dF!Q'o1#43!)[&QA,r5U>qS8
&3b*p\"\\rE,gB'>@:`-4QZG?oo-J@<@k#Q:b5d$3=:$H#\\'Hh\"*\\Zs:Fg(o!4%3ppF
5x#>Bl&>/'4fZFhI_tr,Pd)3(3.O1Zz$y>VU5IG-XY=e6.\\zQ5Nt\\.DzV%)o+Br)yM2I
8F)3\\$o*H&R)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 
0 "" {TEXT -1 89 "Aber nicht immer ist auf ein Computerprogramm Verla
\337! Lassen wir z.B. MAPLE das Integral " }{XPPEDIT 18 0 "s[12];" "6#
&%\"sG6#\"#7" }{TEXT -1 79 " direkt berechnen, so spuckt es eine etwa \+
10 Seiten lange (falsche) Formel aus!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "m:='m':\n'int(sqrt(1+(1+1/2/m)^2*x^(1/m)),x=0..1)'=in
t(sqrt(1+(1+1/12)^2*x^(1/6)),x=0..1):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 269 14 "\nF\374r ungerades" }{TEXT -1 1 " " }{TEXT 267 1 "n" }
{TEXT -1 48 " wird die Berechnung des Integrals\n             " }
{XPPEDIT 18 0 "int(sqrt(1+k^`2`*x^`2/n`),x = a .. b);" "6#-%$intG6$-%%
sqrtG6#,&\"\"\"F**&)%\"kG%\"2GF*)%\"xG%$2/nGF*F*/F0;%\"aG%\"bG" }
{TEXT -1 20 "\nzur Bogenl\344nge von " }{XPPEDIT 18 0 "x^k;" "6#)%\"xG
%\"kG" }{TEXT -1 3 "  (" }{XPPEDIT 18 0 "k = (n+1)/n;" "6#/%\"kG*&,&%
\"nG\"\"\"F(F(F(F'!\"\"" }{TEXT -1 65 " )  leider noch etwas aufwendig
er. Man substituiert  \n          \000" }{XPPEDIT 18 0 "v = sqrt(1+k^`
2`*x^`2/n`);" "6#/%\"vG-%%sqrtG6#,&\"\"\"F)*&)%\"kG%\"2GF))%\"xG%$2/nG
F)F)" }{TEXT -1 26 "  und bekommt\n            " }{XPPEDIT 18 0 "s[n];
" "6#&%\"sG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(sqrt(1+k^`2`
*x^`2/n`),x = 0 .. 1);" "6#-%$intG6$-%%sqrtG6#,&\"\"\"F**&)%\"kG%\"2GF
*)%\"xG%$2/nGF*F*/F0;\"\"!F*" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "n*k^
(-n)*int(v^2*(v^2-1)^(n/2-1),v = 1 .. sqrt(1+k^2));" "6#*(%\"nG\"\"\")
%\"kG,$F$!\"\"F%-%$intG6$*&%\"vG\"\"#),&*$F.F/F%F%F),&*&F$F%F/F)F%F%F)
F%/F.;F%-%%sqrtG6#,&F%F%*$F'F/F%F%" }{TEXT -1 35 " .\nMit einer weiter
en Substitution " }{XPPEDIT 18 0 "v = cosh(t);" "6#/%\"vG-%%coshG6#%\"
tG" }{TEXT -1 86 " gelangt man zum Ziel, aber der Weg ist lang.\nF\374
r die ersten vier ungeraden Werte von " }{TEXT 265 2 "n " }{TEXT -1 
34 "liefert MAPLE folgende Ergebnisse:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 133 "for m from 1 to 4 do print('n'=2*m-1, (2*m-1)*(1+1/(
2*m-1))^(-2*m+1)*int(v^2*(v^2-1)^((2*m-1)/2-1),v=1..sqrt(1+(1+1/(2*m-1
))^2))):od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"nG\"\"\",&*$-%%sqrt
G6#\"\"&F%#F%\"\"#*&#F%\"\"%F%-%#lnG6#,&F-F%F'F%F%F%" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$/%\"nG\"\"$,&#\"$0#\"$G\"\"\"\"*&#\"#\")\"$7&F*-%#ln
G6#F%F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"nG\"\"&,(*$-%%sqrt
G6#\"#h\"\"\"#\"%JM\"&O2#*&#\"&Dc\"\"';W7F,-%#lnG6#F%F,!\"\"*&#F2F3F,-
F56#,&F'F,\"\"'F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"nG\"\"(,(
*$-%%sqrtG6#\"$8\"\"\"\"#\")RmT9\"*'Hj15*&#\")0S#)G\"*caVo#F,-%#lnG6#F
%F,F,*&#F2F3F,-F56#,&F'F,\"\")F,F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "\n\nDie Bogenl\344nge einer kubischen Parabel " }
{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 95 " ist bereits nicht
 mehr mit elementaren Mitteln berechenbar, sondern erfordert ein sogen
anntes " }{TEXT 274 21 "elliptisches Integral" }{TEXT -1 91 ". MAPLE l
iefert leider ohne zus\344tzliche Tricks komplexe Ausdr\374cke f\374r \+
das reelle Integral!?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "1
/3*x*(1+39*x^4)^(1/2)+2/3*1/(3^(1/2)*I)^(1/2)*(1-I*3^(1/2)*x^2)^(1/2)*
(1+3^(1/2)*x^2*I)^(1/2)/(1+3*x^4)^(1/2)*EllipticF(x*(3^(1/2)*I)^(1/2),
I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 970 "K:=(a,b,r)->circle(
[a,b],r,color=black):\nT:=(a1,b1,a2,b2,a3,b3)->polygon([[a1,b1],[a2,b2
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