# Elliptisches Integral  E(xi, k):
> Int(sqrt((1-(3*t/4)^2)/(1-t^2)),t=0..xi)=
> radsimp(int(-sqrt((1-(3*t/4)^2)/(1-t^2)),t=0..xi)assuming xi>0);

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

# Transformation:
> t:=xi*(1+x)/2;

                                xi (1 + x)
                           t := ----------
                                    2

> (xi/2)*Int(sqrt((1-(3*xi*(1+x)/8)^2)/(1-(xi*(1+x)/2)^2)),x=-1..1)=
> EllipticE(xi,3/4); 

               1
              /  /        2        2\1/2
             |   |    9 xi  (1 + x) |
             |   |1 - --------------|
             |   |          64      |
     1/2 xi  |   |------------------|    dx = EllipticE(xi, 3/4)
             |   |       2        2 |
             |   |     xi  (1 + x)  |
             |   | 1 - ------------ |
            /    \          4       /
              -1

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

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

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

  Quadratur[2] :=

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

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

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

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

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

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

  Quadratur[4] :=

                         /          /                1/2 1/2\2\1/2
                         |        2 |    (525 - 70 30   )   | |
                         |    9 xi  |1 - -------------------| |
                         |          \            35         / |
           /        1/2\ |1 - --------------------------------|
           |      30   | |                   64               |
        xi |1/2 + -----| |------------------------------------|
           \       36  / |         /                1/2 1/2\2 |
                         |       2 |    (525 - 70 30   )   |  |
                         |     xi  |1 - -------------------|  |
                         |         \            35         /  |
                         | 1 - ------------------------------ |
                         \                   4                /
        ----------------------------------------------------------
                                    2

                            /          /                1/2 1/2\2\1/2
                            |        2 |    (525 - 70 30   )   | |
                            |    9 xi  |1 + -------------------| |
                            |          \            35         / |
              /        1/2\ |1 - --------------------------------|
              |      30   | |                   64               |
           xi |1/2 + -----| |------------------------------------|
              \       36  / |         /                1/2 1/2\2 |
                            |       2 |    (525 - 70 30   )   |  |
                            |     xi  |1 + -------------------|  |
                            |         \            35         /  |
                            | 1 - ------------------------------ |
                            \                   4                /
         + ----------------------------------------------------------
                                       2

                            /          /                1/2 1/2\2\1/2
                            |        2 |    (525 + 70 30   )   | |
                            |    9 xi  |1 - -------------------| |
                            |          \            35         / |
              /        1/2\ |1 - --------------------------------|
              |      30   | |                   64               |
           xi |1/2 - -----| |------------------------------------|
              \       36  / |         /                1/2 1/2\2 |
                            |       2 |    (525 + 70 30   )   |  |
                            |     xi  |1 - -------------------|  |
                            |         \            35         /  |
                            | 1 - ------------------------------ |
                            \                   4                /
         + ----------------------------------------------------------
                                       2

                            /          /                1/2 1/2\2\1/2
                            |        2 |    (525 + 70 30   )   | |
                            |    9 xi  |1 + -------------------| |
                            |          \            35         / |
              /        1/2\ |1 - --------------------------------|
              |      30   | |                   64               |
           xi |1/2 - -----| |------------------------------------|
              \       36  / |         /                1/2 1/2\2 |
                            |       2 |    (525 + 70 30   )   |  |
                            |     xi  |1 + -------------------|  |
                            |         \            35         /  |
                            | 1 - ------------------------------ |
                            \                   4                /
         + ----------------------------------------------------------
                                       2

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

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

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

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

                      L_zwei[1] := 0.03898791228


                      L_zwei[2] := 0.01585688670


                     L_zwei[3] := 0.008425148396


                     L_zwei[4] := 0.005178056811

# Die Nherungen  mit  n = 3  und  4  GAUSS-Punkten weichen nur im
# Rahmen der Strichstrke vom Elliptischen Integral E(xi,k) ab.
#    Das vollstndige Elliptische Integral  E(1,k)  zweiter Gattung
# ergibt sich zu:
> volstndiges_Ellipt_Integral_zweiter_Gattung:=
> EllipticE(1,3/4)=evalf(EllipticE(1,3/4));

  volstndiges_Ellipt_Integral_zweiter_Gattung :=

        EllipticE(3/4) = 1.318472108

#    Das vollstndige Elliptische Integral F(1,k) erster Gattung erhlt
# man entsprechend:                                                     
#                       
> vollstndiges_Ellipt_Integral_erster_Gattung:=
> EllipticF(1,3/4)=evalf(EllipticF(1,3/4));

  vollstndiges_Ellipt_Integral_erster_Gattung :=

        EllipticK(3/4) = 1.910989781

> 
> 
