> Int(1/sqrt((1-t^2)*(1-(3*t/4)^2)),t=0..xi)=
> radsimp(int(1/sqrt((1-t^2)*(1-(3*t/4)^2)),t=0..xi)assuming xi>0);

           xi
          /
         |               1
         |    ------------------------ dt = EllipticF(xi, 3/4)
         |    /         /       2\\1/2
         |    |      2  |    9 t ||
        /     |(1 - t ) |1 - ----||
          0   \         \     16 //

> (xi/2)*Int(1/sqrt((1-(xi*(1+x)/2)^2)*(1-((3/4)*xi*(1+x)/2)^2)),x=-1..1
> )= EllipticF(xi,3/4); 
> 

            1
           /
          |                        1
  1/2 xi  |   -------------------------------------------- dx =
          |   //      2        2\ /        2        2\\1/2
          |   ||    xi  (1 + x) | |    9 xi  (1 + x) ||
         /    ||1 - ------------| |1 - --------------||
           -1 \\         4      / \          64      //

        EllipticF(xi, 3/4)

> Quadratur[1]:=xi/sqrt((1-xi^2/4)*(1-9*xi^2/64));

                                          xi
              Quadratur[1] := --------------------------
                              //      2\ /        2\\1/2
                              ||    xi | |    9 xi ||
                              ||1 - ---| |1 - -----||
                              \\     4 / \     64  //

> Quadratur[2]:=(xi/2)/sqrt((1-xi^2*(1-1/sqrt(3))^2/4)*
> (1-9*xi^2*(1-1/sqrt(3))^2/64))+(xi/2)/sqrt((1-xi^2*(1+1/sqrt(3))^2/4)*
> (1-9*xi^2*(1+1/sqrt(3))^2/64));
> 

  Quadratur[2] :=

                                 xi
        ----------------------------------------------------
          //        /     1/2\2\ /          /     1/2\2\\1/2
          ||      2 |    3   | | |        2 |    3   | ||
          ||    xi  |1 - ----| | |    9 xi  |1 - ----| ||
          ||        \     3  / | |          \     3  / ||
        2 ||1 - ---------------| |1 - -----------------||
          \\           4       / \           64        //

                                    xi
         + ----------------------------------------------------
             //        /     1/2\2\ /          /     1/2\2\\1/2
             ||      2 |    3   | | |        2 |    3   | ||
             ||    xi  |1 + ----| | |    9 xi  |1 + ----| ||
             ||        \     3  / | |          \     3  / ||
           2 ||1 - ---------------| |1 - -----------------||
             \\           4       / \           64        //

> Quadratur[3]:=(xi/2)*(8/9)/sqrt((1-xi^2/4)*(1-9*xi^2/64))+(xi/2)*(5/9)
> /sqrt((1-xi^2*(1-sqrt(3/5))^2/4)*(1-9*xi^2*(1-sqrt(3/5))^2/64))+(xi/2)
> *(5/9)/sqrt((1-xi^2*(1+sqrt(3/5))^2/4)*(1-9*xi^2*(1+sqrt(3/5))^2/64));

                              4 xi
  Quadratur[3] := ----------------------------
                    //      2\ /        2\\1/2
                    ||    xi | |    9 xi ||
                  9 ||1 - ---| |1 - -----||
                    \\     4 / \     64  //

                                    5 xi
         + -------------------------------------------------------
              //        /      1/2\2\ /          /      1/2\2\\1/2
              ||      2 |    15   | | |        2 |    15   | ||
              ||    xi  |1 - -----| | |    9 xi  |1 - -----| ||
              ||        \      5  / | |          \      5  / ||
           18 ||1 - ----------------| |1 - ------------------||
              \\           4        / \            64        //

                                    5 xi
         + -------------------------------------------------------
              //        /      1/2\2\ /          /      1/2\2\\1/2
              ||      2 |    15   | | |        2 |    15   | ||
              ||    xi  |1 + -----| | |    9 xi  |1 + -----| ||
              ||        \      5  / | |          \      5  / ||
           18 ||1 - ----------------| |1 - ------------------||
              \\           4        / \            64        //

> n:=4:
> g:=sqrt(525-70*sqrt(30))/35; G:=sqrt(525+70*sqrt(30))/35;    
> H:=1/2+sqrt(30)/36; h:=1/2-sqrt(30)/36;

                                        1/2 1/2
                            (525 - 70 30   )
                       g := -------------------
                                    35


                                        1/2 1/2
                            (525 + 70 30   )
                       G := -------------------
                                    35


                                        1/2
                                      30
                           H := 1/2 + -----
                                       36


                                        1/2
                                      30
                           h := 1/2 - -----
                                       36

> Quadratur[4]:=(xi/2)*H/sqrt((1-xi^2*(1-g)^2/4)*(1-9*xi^2*(1-g)^2/64))+
> (xi/2)*H/sqrt((1-xi^2*(1+g)^2/4)*(1-9*xi^2*(1+g)^2/64))+(xi/2)*h/sqrt(
> (1-xi^2*(1-G)^2/4)*(1-9*xi^2*(1-G)^2/64))+(xi/2)*h/sqrt((1-xi^2*(1+G)^
> 2/4)*(1-9*xi^2*(1+G)^2/64));

                                       /  /
                                       |  |
                     /        1/2\     |  |
                     |      30   |   / |  |
  Quadratur[4] := xi |1/2 + -----|  /  |2 |
                     \       36  / /   \  \

        /        /                1/2 1/2\2\
        |      2 |    (525 - 70 30   )   | |
        |    xi  |1 - -------------------| |
        |        \            35         / |
        |1 - ------------------------------|
        \                  4               /

        /          /                1/2 1/2\2\\1/2\
        |        2 |    (525 - 70 30   )   | ||   |
        |    9 xi  |1 - -------------------| ||   |
        |          \            35         / ||   |
        |1 - --------------------------------||   | + xi
        \                   64               //   /

                          /  //        /                1/2 1/2\2\
                          |  ||      2 |    (525 - 70 30   )   | |
        /        1/2\     |  ||    xi  |1 + -------------------| |
        |      30   |   / |  ||        \            35         / |
        |1/2 + -----|  /  |2 ||1 - ------------------------------|
        \       36  / /   \  \\                  4               /

        /          /                1/2 1/2\2\\1/2\
        |        2 |    (525 - 70 30   )   | ||   |
        |    9 xi  |1 + -------------------| ||   |
        |          \            35         / ||   |
        |1 - --------------------------------||   | + xi
        \                   64               //   /

                          /  //        /                1/2 1/2\2\
                          |  ||      2 |    (525 + 70 30   )   | |
        /        1/2\     |  ||    xi  |1 - -------------------| |
        |      30   |   / |  ||        \            35         / |
        |1/2 - -----|  /  |2 ||1 - ------------------------------|
        \       36  / /   \  \\                  4               /

        /          /                1/2 1/2\2\\1/2\
        |        2 |    (525 + 70 30   )   | ||   |
        |    9 xi  |1 - -------------------| ||   |
        |          \            35         / ||   |
        |1 - --------------------------------||   | + xi
        \                   64               //   /

                          /  //        /                1/2 1/2\2\
                          |  ||      2 |    (525 + 70 30   )   | |
        /        1/2\     |  ||    xi  |1 + -------------------| |
        |      30   |   / |  ||        \            35         / |
        |1/2 - -----|  /  |2 ||1 - ------------------------------|
        \       36  / /   \  \\                  4               /

        /          /                1/2 1/2\2\\1/2\
        |        2 |    (525 + 70 30   )   | ||   |
        |    9 xi  |1 + -------------------| ||   |
        |          \            35         / ||   |
        |1 - --------------------------------||   |
        \                   64               //   /

> plot({EllipticF(1,3/4),EllipticF(xi,3/4),
> Quadratur[1],Quadratur[2],Quadratur[3],Quadratur[4]},xi=0..1,color=bla
> ck);

> L_zwei:=sqrt(Int((Ellipt_Integral[1]-Quadratur)^2,xi=0..1));

                /   1                                      \1/2
                |  /                                       |
                | |                                   2    |
      L_zwei := | |   (Ellipt_Integral[1] - Quadratur)  dxi|
                | |                                        |
                |/                                         |
                \  0                                       /

# 
> for i in [1,2,3,4] do 
> L_zwei[i]:=evalf(sqrt(int((EllipticF(xi,3/4)-Quadratur[i])^2,xi=0..1))
> )            od;

                      L_zwei[1] := 0.1148834294


                      L_zwei[2] := 0.04045580767


                      L_zwei[3] := 0.02028975381


                      L_zwei[4] := 0.01218529772

#    Funktionswerte an der Stelle  xi = 1
>                                                                       
>        for i in [1,2,3,4] do Q[i](1):= evalf(subs(xi=1,Quadratur[i]))
> od;
> 

                        Q[1](1) := 1.245598311


                        Q[2](1) := 1.526732352


                        Q[3](1) := 1.641077696


                        Q[4](1) := 1.702461182

> vollstndiges_Ellipt_Integral_erster_Gattung:=EllipticF(1,3/4);

    vollstndiges_Ellipt_Integral_erster_Gattung := EllipticK(3/4)

> vollstndiges_Ellipt_Integral_erster_Gattung:=evalf(%);

     vollstndiges_Ellipt_Integral_erster_Gattung := 1.910989781

# Die Nherungen mit  n = 2, 3 und 4 GAUSS-Punkten sind im gesamten
# Bereich  < 0..1  >  zufrieden stellend. 
#    Betrachtet man den um 1% verkrzten Bereich  < 0..0.99 >  , so
# ergeben sich wesentlich geringere L-zwei-Werte,
# wie nachstehender Ausdruck zeigt.
> for i in [1,2,3,4] do                              
> L_zwei[i]:=evalf(sqrt(int((EllipticF(xi,3/4)-Quadratur[i])^2,xi=0..0.9
> 9)))  od;
# 
#                       L_zwei[1] := 0.1017016032
# 

                      L_zwei[1] := 0.1017016032


                      L_zwei[2] := 0.03069055101


                      L_zwei[3] := 0.01265839741


                     L_zwei[4] := 0.006073938879

> 
# Ergnzungen zu den Elliptischen Integralen erster und zweiter Gattung,
# F(xi,k) und E(xi,k):
> Int(1/sqrt((1-t^2)*(1-k^2*t^2)),t=0..xi)=
> radsimp(-int(1/sqrt((1-t^2)*(1-k^2*t^2)),t=0..xi) assuming k>0);

           xi
          /
         |                1
         |    ------------------------- dt = EllipticF(xi, k)
         |           2        2  2  1/2
        /     ((1 - t ) (1 - k  t ))
          0

> plot({2,EllipticF(xi,0),EllipticF(xi,1/4),EllipticF(xi,1/2),
> EllipticF(xi,3/4),EllipticF(xi,1)},xi=0..1,0..2,color=black);

> Grenzwert[1][0]:=limit(EllipticF(xi,0),xi=1);

                                           Pi
                       Grenzwert[1][0] := ----
                                           2

> Grenzwert[1][1]:=limit(EllipticF(xi,1),xi=1);

                     Grenzwert[1][1] := infinity

> Int(sqrt((1-k^2*t^2)/(1-t^2)),t=0..xi)=
> radsimp(-int(sqrt((1-k^2*t^2)/(1-t^2)),t=0..xi) assuming k>0);

                 xi
                /   /     2  2\1/2
               |    |1 - k  t |
               |    |---------|    dt = EllipticE(xi, k)
               |    |      2  |
              /     \ 1 - t   /
                0

> plot({EllipticE(1,0),EllipticE(xi,0),EllipticE(xi,1/4),
> EllipticE(xi,1/2),EllipticE(xi,3/4),EllipticE(xi,1)},
> xi=0..1,0..evalf(EllipticE(1,0)),color=black);

> 
> Grenzwert[2][0]:=limit(EllipticE(xi,0),xi=1);

                                           Pi
                       Grenzwert[2][0] := ----
                                           2

> Grenzwert[2][1]:=limit(EllipticE(xi,1),xi=1);

                         Grenzwert[2][1] := 1

> 
