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Approximations of Elliptic Integrals Univ.-Prof. Dr.-Ing. habil. J. BETTEN RWTH University Aachen Mathematical Models in Materials Science and Continuum Mechanics Augustinerbach 4-20 D-52056 A a c h e n , Germany betten@mmw.rwth-aachen.de

Using MAPLE V , Release 10, Elliptic Integrals hve been presented. Furthermore, the author proposes approximations based on both FOURIER and TAYLOR series. Keywords: Elliptic Integrals; Approximation; FOURIER-Series; TAYLOR-Series

restart: E(t,k):=Int(sqrt(1-k^2*(sin(x))^2),x=0..arcsin(t))=simplify(int(sqrt(1-k^2*(sin(x))^2),x=0..arcsin(t))) assuming t>=0 and t<1, k>=0 and k<1;

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiQsJiIiIkYsKiZJImtHRigiIiMtSSRzaW5HRiU2I0kieEdGKEYvISIiI0YsRi8vRjM7IiIhLUknYXJjc2luR0YlNiNJInRHRigtSSpFbGxpcHRpY0VHRiU2JEY8Ri4=

for i in [0,1/2,2/3,1] do E(t,i):=radsimp(subs(k=i,rhs(E(t,k)))) od;

LUknYXJjc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInRHRic=

LUkqRWxsaXB0aWNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJInRHRicjIiIiIiIj

LUkqRWxsaXB0aWNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJInRHRicjIiIjIiIk

SSJ0RzYi

E(1,k):=EllipticE(1,k);

LUkqRWxsaXB0aWNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJImtHRic=

for i in [0,1/2,2/3,1] do E(1,i):= evalf(subs(k=i,EllipticE(k))) od;

JCIrRmp6cTohIio=

JCIrNEFZbjkhIio=

JCIrUVI1eTghIio=

JCIiIiIiIQ==

alias(H=Heaviside,th=thickness,co=color): plot1:=plot({E(t,0),E(t,1/2),E(t,2/3),E(t,1)},t=0..1,scaling=constrained,th=1,co=black): plot2:=plot({1,1.3781,1.4675,Pi/2,(Pi/2)*H(t-1)},t=0..1.001,co=black): plots[display](plot1,plot2);

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for i in [0,1/4,1/2,3/4,1] do E(i,2/3):= evalf(simplify(subs(t=i,E(t,2/3)))) od;

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JCIrXFMmXF4jISM1

JCIrKCkqZU44JiEjNQ==

JCIrL25hcyEpISM1

XiQkIitRUjV5OCEiKiQhIiEiIiE=

alias(H=Heaviside,th=thickness,c=color): plot1:=plot(E(t,2/3),t=0..1,scaling=constrained,th=1,c=black): plot2:=plot({E(1,2/3),E(1,2/3)*H(t-1)},t=0..1.001,c=black): plots[display](plot1,plot2);

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NXk4ISIqNyQkIi52YG4lPSIqbyEjOCQiK1FSNXk4ISIqNyQkIjEjZTltbFFxMyghIzskIitRUjV5OCEiKjckJCIuRCoqeWU2SSghIzgkIitRUjV5OCEiKjckJCIxOnpmWyJIRV0oISM7JCIrUVI1eTghIio3JCQiMVtpWENQQTh4ISM7JCIrUVI1eTghIio3JCQiMWw7YTolPSI+eiEjOyQiK1FSNXk4ISIqNyQkIjFbN3l2NW5NIikhIzskIitRUjV5OCEiKjckJCIxTExqMVJGVSQpISM7JCIrUVI1eTghIio3JCQiMUszI2Z5eFhiKSEjOyQiK1FSNXk4ISIqNyQkIjE7eio9b0JedykhIzskIitRUjV5OCEiKjckJCIxKioqXFFcJGZlKikhIzskIitRUjV5OCEiKjckJCIxbFRTSjxMIT0qISM7JCIrUVI1eTghIio3JCQiMUpMVmFxbHkkKiEjOyQiK1FSNXk4ISIqNyQkIjFbNyRSQDgsZiohIzskIitRUjV5OCEiKjckJCIxXChvKFJQXSN6KiEjOyQiK1FSNXk4ISIqNyQkIiUsNSEiJCQiK1FSNXk4ISIqLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSVWSUVXRzYkOyQiIiEhIiIkIiUsNSEiJCUoREVGQVVMVEctJSpBWEVTU1RZTEVHNiMlJ05PUk1BTEctJShTQ0FMSU5HRzYjJSxDT05TVFJBSU5FREctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiU1RCEiIi0lKUJPVU5EU19ZRzYjJCIjISkhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVnPSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUhZiMhIiItJSlDSElMRFJFTkc2Ig==

Integrand w(x) des elliptischen Integrals E(t, 2/3) zweiter Gattung in der LEGENDREschen Normalform f\303\274r k = 2/3: w(x):=sqrt(1-(4/9)*(sin(x))^2);

LCQqJCwmIiIqIiIiKiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGLSIiIyEiJSNGJkYwI0YmIiIk

w(Pi/2):=simplify(subs(x=Pi/2,%));

LCQqJCIiJiMiIiIiIiMjRiYiIiQ=

w(Pi/2):=evalf(%);

JCIrQipmTlgoISM1

Der Integrand w(x) wird angn\303\244hert durch folgende FOURIER-Reihe, die weiter unten hergeleitet wird: F(x):=0.87733+0.12698*cos(2*x)-0.0046192*cos(4*x);

LCgkIiZMeCkhIiYiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ4R0YsIiIjJCImKXA3RiUtRig2IywkRi8iIiUkISYjPlkhIig=

plot3:=plot(w(x),x=0..2*Pi,sqrt(5)/3..1,th=1,c=red): plot4:=plot(F(x),x=0..2*Pi,th=3,c=blue,transparency=0.6,linestyle=DASH): plot5:=plot({1,H(x-2*Pi)},x=0..2.001*Pi,c=black): plots[display](plot3,plot4,plot5);

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Herleitung der FOURIER-Reihe F(x) des Integranden w(x) des elliptischen Integrals E(t, k): restart: Fourierreihe:= a[0]/2+sum(a[k]*cos(k*x)+b[k]*sin(k*x),k=1..infinity);

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a[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=-Pi..Pi); # k=0,1,2,...

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a[0]:=simplify(subs(k=0,%));

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b[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=-Pi..Pi); # k=0,1,2,...

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Integrand des elliptischen Integrals: w(x):=sqrt(1-(4/9)*(sin(x))^2);

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A[0]:=value(subs(f(x)=w(x),a[0]));

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A[0]:=evalf(%,6);

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A[k]:=value(subs(f(x)=w(x),a[k]));

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for i in [0,2,4,6] do A[i]:=evalf(value(subs(k=i,A[k])),6) od;

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B[k]:=value(subs(f(x)=w(x),b[k]));

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Da der Integrand eine gerade Funktion ist, stellt die FOURIER-Reihe eine Cosinus-Reihe dar: Fourierreihe[k=4]:=A[0]/2+sum(A[2*k]*cos(2*k*x),k=1..2);

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F(x):=evalf(%,5);

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for i in [0,Pi/2,Pi] do F(i):=evalf(simplify(subs(x=i,F(x))),5) od;

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Die exakten Werte des Integranden sind F(0) = F(Pi) = 1 und F(Pi/2) = 0.745356. Das Elliptische Integral E(t, 2/3) wird ersetzt durch die N\303\244herung e(t, 2/3) , die man durch folgende Integration erh\303\244lt: e(t,2/3):=Int(F(xi),xi=0..arcsin(t))=int(F(x),x=0..arcsin(t));

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e(t,2/3):=evalf(rhs(%),5);

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Diese Funktion kann auch folgenderma\303\237en ausgedr\303\274ckt werden: e_(t,2/3):=0.87733*expand(arcsin(t))+ 0.06349*expand(sin(2*arcsin(t)))- 0.0011548*expand(sin(4*arcsin(t)));

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for i in [0,1/4,1/2,3/4,1] do e(i,2/3):=evalf(subs(t=i,e(t,2/3))) od;

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for i in [0,1/4,1/2,3/4,1] do e_(i,2/3):=evalf(subs(t=i,e_(t,2/3))) od;

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E(t,2/3):=EllipticE(t,2/3);

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for i in [0,1/4,1/2,3/4,1] do E(i,2/3):=evalf(subs(t=i,E(t,2/3))) od;

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L-zwei Fehlernorm des Integranden: L[2]:=sqrt((1/T)*Int((w(tau)-F(tau))^2,tau=0..T));

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for i in [Pi/2,Pi] do L_zwei[0..i]:= evalf(subs(t=i,sqrt((1/i)*int((w(x)-F(x))^2,x=0..t))),5) od;

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JCImJypRIyEiKQ==

Die Werte zeigen, dass F(x) eine gute N\303\244herung des Integranden w(x) darstellt. alias(H=Heaviside,th=thickness,c=color): plot1:=plot(E(t,2/3),t=0..1,scaling=constrained,th=5,c=red,transparency=0.6): plot2:=plot(e(t,2/3),t=0..1,scaling=constrained,th=1,c=black): plot3:=plot({1.378,1.378*H(t-1)},t=0..1.001,c=black): plots[display](plot1,plot2,plot3);

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IiV5OCEiJDckJCIxKioqXHl0O1okZSEjOyQiJXk4ISIkNyQkIjEqXGl4a0JXLychIzskIiV5OCEiJDckJCIxXDcubWVLWmkhIzskIiV5OCEiJDckJCIxKlxQYj05RVonISM7JCIleTghIiQ3JCQiMWxtcmdkL3ZtISM7JCIleTghIiQ3JCQiLnZgbiU9IipvISM4JCIleTghIiQ3JCQiMSNlOW1sUXEzKCEjOyQiJXk4ISIkNyQkIi5EKip5ZTZJKCEjOCQiJXk4ISIkNyQkIjE6emZbIkhFXSghIzskIiV5OCEiJDckJCIxW2lYQ1BBOHghIzskIiV5OCEiJDckJCIxbDthOiU9Ij56ISM7JCIleTghIiQ3JCQiMVs3eXY1bk0iKSEjOyQiJXk4ISIkNyQkIjFMTGoxUkZVJCkhIzskIiV5OCEiJDckJCIxSzMjZnl4WGIpISM7JCIleTghIiQ3JCQiMTt6Kj1vQl53KSEjOyQiJXk4ISIkNyQkIjEqKipcUVwkZmUqKSEjOyQiJXk4ISIkNyQkIjFsVFNKPEwhPSohIzskIiV5OCEiJDckJCIxSkxWYXFseSQqISM7JCIleTghIiQ3JCQiMVs3JFJAOCxmKiEjOyQiJXk4ISIkNyQkIjFcKG8oUlBdI3oqISM7JCIleTghIiQ3JCQiJSw1ISIkJCIleTghIiQtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lJ0NVUlZFU0c2JDdnbjckJCIiISEiIiQiIiEhIiI3JCQiMUxMM0teKj09IyEjPCQiIiEhIiI3JCQiMW0ieio+L04hMyUhIzwkIiIhISIiNyQkIjFLJGVXJT5POmkhIzwkIiIhISIiNyQkIjFMJDM3ckRYTykhIzwkIiIhISIiNyQkIjJrIkh4KFtaLjAiISM8JCIiISEiIjckJCIySmVrZlhiJ1s3ISM8JCIiISEiIjckJCIyJ1w3ZXE9KlJYIiEjPCQiIiEhIiI3JCQiMktlOVYzXmptIiEjPCQiIiEhIiI3JCQiMilcaU5mI0gheT0hIzwkIiIhISIiNyQkIjJqbVQ0R21kNCMhIzwkIiIhISIiNyQkIjFMMzJvIlx2RyMhIzskIiIhISIiNyQkIi5EYEZeTV0jISM4JCIiISEiIjckJCIyKCoqXFAoKilSLXMjISM8JCIiISEiIjckJCIudlkhZjpISCEjOCQiIiEhIiI3JCQiMSRlOVFscik9SiEjOyQiIiEhIiI3JCQiMW07Q0g/WVdMISM7JCIiISEiIjckJCIxbW0nem1tYmAkISM7JCIiISEiIjckJCIyJFw3JFJ6bXl2JCEjPCQiIiEhIiI3JCQiMW1tcnMra2FSISM7JCIiISEiIjckJCIwREprYUowPCUhIzokIiIhISIiNyQkIjIkXFBIJEc2aFAlISM8JCIiISEiIjckJCIybDt6Pjw3MWYlISM8JCIiISEiIjckJCIxO0hzWjRmKHklISM7JCIiISEiIjckJCIxTDNGbyJmKysmISM7JCIiISEiIjckJCIxJGU5aiRSdj9fISM7JCIiISEiIjckJCIxXChvMVFwR1QmISM7JCIiISEiIjckJCIxTGV1QiRmLmkmISM7JCIiISEiIjckJCIxKioqXHl0O1okZSEjOyQiIiEhIiI3JCQiMSpcaXhrQlcvJyEjOyQiIiEhIiI3JCQiMVw3Lm1lS1ppISM7JCIiISEiIjckJCIxKlxQYj05RVonISM7JCIiISEiIjckJCIxbG1yZ2Qvdm0hIzskIiIhISIiNyQkIi52YG4lPSIqbyEjOCQiIiEhIiI3JCQiMSNlOW1sUXEzKCEjOyQiIiEhIiI3JCQiLkQqKnllNkkoISM4JCIiISEiIjckJCIxOnpmWyJIRV0oISM7JCIiISEiIjckJCIxW2lYQ1BBOHghIzskIiIhISIiNyQkIjFsO2E6JT0iPnohIzskIiIhISIiNyQkIjFbN3l2NW5NIikhIzskIiIhISIiNyQkIjFMTGoxUkZVJCkhIzskIiIhISIiNyQkIjFLMyNmeXhYYikhIzskIiIhISIiNyQkIjE7eio9b0JedykhIzskIiIhISIiNyQkIjEqKipcUVwkZmUqKSEjOyQiIiEhIiI3JCQiMWxUU0o8TCE9KiEjOyQiIiEhIiI3JCQiMUpMVmFxbHkkKiEjOyQiIiEhIiI3JCQiMVs3JFJAOCxmKiEjOyQiIiEhIiI3JCQiMSkqKlxvWjM4cCohIzskIiIhISIiNyQkIjFcKG8oUlBdI3oqISM7JCIiISEiIjckJCIxaGwjWyF5KG8lKSohIzskIiIhISIiNyQkIjF0VikpcD1ELCoqISM7JCIiISEiIjckJCIxektULSpRJUcqKiEjOyQiIiEhIiI3JCQiMSc9VVwkZmliKiohIzskIiIhISIiNyQkIjFSbT9eJT4jcCoqISM7JCIiISEiIjckJCIxJDRydSdIIkcpKiohIzskIiIhISIiNyQkIjE+TGdEKDQnKikqKiEjOyQiIiEhIiI3JCQiMVlidCRbMWsqKiohIzskIiIhISIiNyQkIjJ0eCc9Qy5LKzUhIzskIiV5OCEiJDckJCIlLDUhIiQkIiV5OCEiJC0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUlVklFV0c2JDskIiIhISIiJCIlLDUhIiQlKERFRkFVTFRHLSUqQVhFU1NUWUxFRzYjJSdOT1JNQUxHLSUoU0NBTElOR0c2IyUsQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlP0shIiItJSlCT1VORFNfWUc2IyQiJCsiISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlPz4hIiItJS5CT1VORFNfSEVJR0hURzYjJCIlIW8jISIiLSUpQ0hJTERSRU5HNiI=

Die N\303\244herung e(t, 2/3) unterscheidet sich vom Elliptischen Integral E(t, 2/3) nur innerhalb der Strichst\303\244rke und ist somit genauer und wesentlich einfacher als die in [BETTEN, J.: Finite Elemente f\303\274r Ingenieure 2, zweite Auflage, Springer-Verlag, 2004, * 7.5-23.mws ] diskutierte GAUSS-Quadratur! Wahrer Fehler f\303\274r k = 2/3: Delta(t,2/3):=E(t,2/3)-e(t,2/3);

LCotSSpFbGxpcHRpY0VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkidEdGKCMiIiMiIiQiIiItSSdhcmNzaW5HRiU2I0YqJCEmTHgpISImLUkkc2luR0YlNiMsJEYvJEYsIiIhJCEmIVxqISInLUY2NiMsJEYvJCIiJUY6JCImWzoiISIo

plot(Delta(t,2/3),t=0..1,c=black);

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

Relativer Fehler f\303\274r k = 2/3: relativer_Fehler(t,2/3):=Delta(t,2/3)/E(t,2/3);

KiYsKi1JKkVsbGlwdGljRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJ0R0YpIyIiIyIiJCIiIi1JJ2FyY3NpbkdGJjYjRiskISZMeCkhIiYtSSRzaW5HRiY2IywkRjAkRi0iIiEkISYhXGohIictRjc2IywkRjAkIiIlRjskIiZbOiIhIihGL0YkISIi

plot(relativer_Fehler(t,2/3),t=0..1,c=black);

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

relativer_Fehler(0,2/3):=Limit(rel_Fehler(t,2/3),t=0)= evalf(limit(relativer_Fehler(t,2/3),t=0),4);

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSStyZWxfRmVobGVyR0YoNiRJInRHRigjIiIjIiIkL0YtIiIhJCIlIzQkISIo

Bemerkung: Der wahre Fehler Delta(t, 2/3) := E(t, 2/3) - e(t, 2/3) in [0, 1] liegt zwischen -0.00005 und 0.00006 und der relative Fehler zwischen -0.00005 und 0.0003. Mithin ist die auf der Basis einer FOURIER-Analyse gewonnene N\303\244herung sehr gelungen. Approximationen elliptischer Integrale erster und dritter Gattung oder auch Integrale, die von RAMANUJAN untersucht wurden, k\303\266nnen in gleicher Weise approximiert werden.

Im Folgenden wird das elliptische Integral E(t, 2/3) durch eine TAYLOR-Reihe gen\303\244hert: restart: w(x):=sqrt(1-(2/3)^2*(sin(x))^2);

LCQqJCwmIiIqIiIiKiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGLSIiIyEiJSNGJkYwI0YmIiIk

TAYLOR_Reihe(x):=taylor(w(x),x=0,5);

KytJInhHNiIiIiIiIiEjISIjIiIqIiIjIyIiJSIjIilGLC1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

T[4](x):=1-2*x^2/9+4*x^4/81;

LCgiIiJGIyokSSJ4RzYiIiIjIyEiIyIiKiokRiUiIiUjRiwiIyIp

e(t,2/3):=Int(TAYLOR[4](x),x=0..arcsin(t))= int(T[4](x),x=0..arcsin(t));

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJJ1RBWUxPUkdGKDYjIiIlNiNJInhHRigvRjA7IiIhLUknYXJjc2luR0YlNiNJInRHRigsKEY0IiIiKiRGNCIiJCMhIiMiI0YqJEY0IiImI0YuIiQwJQ==

e(1,2/3):=simplify(subs(t=1,%));

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJJ1RBWUxPUkdGKDYjIiIlNiNJInhHRigvRjA7IiIhLUknYXJjc2luR0YlNiMiIiIsKEkjUGlHRiYjRjciIiMqJEY0IiIkIyEiIyIjRiokRjQiIiYjRi4iJDAl

e(1,2/3):=evalf(%);

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLSZJJ1RBWUxPUkdGKDYjIiIlNiNJInhHRigvRjA7JCIiIUY0JCIrRmp6cTohIiokIisjcF4ieThGNw==

E(t,2/3):=EllipticE(t,2/3);

LUkqRWxsaXB0aWNFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJInRHRicjIiIjIiIk

E(1,2/3):=evalf(subs(t=1,%));

XiQkIitRUjV5OCEiKiQhIiEiIiE=

alias(H=Heaviside,th=thickness,c=color): plot1:=plot(E(t,2/3),t=0..1,th=1,c=blue): plot2:=plot(rhs(e(t,2/3)),t=0..1,th=3,c=red,transparency=0.5,linestyle=DASH): plot3:=plot({E(1,2/3),E(1,2/3)*H(t-1)}, t=0..1.001,scaling=constrained,c=black): plots[display](plot1,plot2,plot3);

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wahrer_Fehler(t,2/3):=E(t,2/3)-rhs(e(t,2/3));

LCotSSpFbGxpcHRpY0VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkidEdGKCMiIiMiIiQiIiItSSdhcmNzaW5HRiU2I0YqISIiKiRGL0YtI0YsIiNGKiRGLyIiJiMhIiUiJDAl

wahrer_Fehler(1,2/3):=evalf(subs(t=1,%));

XiQkIShZYXglISM2JCEiISIiIQ==

plot(wahrer_Fehler(t,2/3),t=0..1,th=2,c=black);

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

relativer_Fehler(t,2/3):= wahrer_Fehler(t,2/3)/E(t,2/3);

KiYsKi1JKkVsbGlwdGljRUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJ0R0YpIyIiIyIiJCIiIi1JJ2FyY3NpbkdGJjYjRishIiIqJEYwRi4jRi0iI0YqJEYwIiImIyEiJSIkMCVGL0YkRjM=

relativer_Fehler(1,2/3):=evalf(subs(t=1,%));

XiQkISsmPkhfWSQhIzkkISIhIiIh

relativer_Fehler(0,2/3):=Limit(rel_Fehler(t,2/3),t=0)= limit(relativer_Fehler(t,2/3),t=0);

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSStyZWxfRmVobGVyR0YoNiRJInRHRigjIiIjIiIkL0YtIiIhRjI=

plot(relativer_Fehler(t,2/3),t=0..1,th=2,c=black);

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

Die G\303\274te der Approximation des elliptischen Integrals E(t, 2/3) durch eine TAYLOR-Reihe ist vergleichbar mit der G\303\274te der N\303\244herung auf der Basis einer FOURIER-Analyse.

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