#  Dilogarithmus nach EULER
> R(t):=ln(t)/(1-t);Int(R(t),t=1..xi)=int(R(t),t=1..xi);
> 

                                    ln(t)
                            R(t) := -----
                                    1 - t


                         xi
                        /
                       |    ln(t)
                       |    ----- dt = dilog(xi)
                       |    1 - t
                      /
                        1

> t(x,xi):=(1+xi)/2-(1-xi)*x/2;

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

> j:=diff(t(x,xi),x);

                                        xi
                          j := - 1/2 + ----
                                        2

> r(x,xi):=subs(t=t(x,xi),R(t));

                                     xi    (1 - xi) x
                           ln(1/2 + ---- - ----------)
                                     2         2
               r(x, xi) := ---------------------------
                                    xi    (1 - xi) x
                             1/2 - ---- + ----------
                                    2         2

> j*Int(r(x,xi),x=-1..1)=simplify(j*int(r(x,xi),x=-1..1));
> 
> 

                      1
                     /            xi    (1 - xi) x
                    |   ln(1/2 + ---- - ----------)
    /         xi \  |             2         2
    |- 1/2 + ----|  |   --------------------------- dx = dilog(xi)
    \         2  /  |            xi    (1 - xi) x
                    |     1/2 - ---- + ----------
                   /             2         2
                     -1

> plot({0,dilog(0),dilog(5),dilog(xi)},xi=0..5,
> dilog(0)..dilog(5),scaling=constrained,axes=boxed, color=black);
# 

> n:=1:  r[1]:=subs(x=0,r(x,xi));

                                          xi
                                ln(1/2 + ----)
                                          2
                        r[1] := --------------
                                         xi
                                  1/2 - ----
                                         2

> Quadratur[1]:=2*j*%;

                             /         xi \           xi
                           2 |- 1/2 + ----| ln(1/2 + ----)
                             \         2  /           2
           Quadratur[1] := -------------------------------
                                            xi
                                     1/2 - ----
                                            2

> quadratur[1]:=simplify(%);

                quadratur[1] := 2 ln(2) - 2 ln(1 + xi)

> 
> n:=2:
> r[2][1]:=ln(1/2+xi/2+(1-xi)/sqrt(3)/2)/(1/2-xi/2-(1-xi)/sqrt(3)/2);
> 

                                                   1/2
                                   xi    (1 - xi) 3
                         ln(1/2 + ---- + -------------)
                                   2           6
              r[2][1] := ------------------------------
                                                  1/2
                                  xi    (1 - xi) 3
                           1/2 - ---- - -------------
                                  2           6

> r[2][2]:=ln(1/2+xi/2-(1-xi)/sqrt(3)/2)/(1/2-xi/2+(1-xi)/sqrt(3)/2);

                                                   1/2
                                   xi    (1 - xi) 3
                         ln(1/2 + ---- - -------------)
                                   2           6
              r[2][2] := ------------------------------
                                                  1/2
                                  xi    (1 - xi) 3
                           1/2 - ---- + -------------
                                  2           6

> Quadratur[2]:=j*(r[2][1]+r[2][2]);

                                 /                          1/2
                                 |          xi    (1 - xi) 3
                                 |ln(1/2 + ---- + -------------)
                  /         xi \ |          2           6
  Quadratur[2] := |- 1/2 + ----| |------------------------------
                  \         2  / |                         1/2
                                 |         xi    (1 - xi) 3
                                 |  1/2 - ---- - -------------
                                 \         2           6

                                     1/2 \
                     xi    (1 - xi) 3    |
           ln(1/2 + ---- - -------------)|
                     2           6       |
         + ------------------------------|
                                    1/2  |
                    xi    (1 - xi) 3     |
             1/2 - ---- + -------------  |
                    2           6        /

> quadratur[2]:=simplify(%);

                               1/2   1/2                    1/2   1/2
  quadratur[2] := -1/2 ln(3 - 3   ) 3    - 1/2 ln(2 + xi + 3   ) 3

                       1/2   1/2                    1/2   1/2
         + 1/2 ln(3 + 3   ) 3    + 1/2 ln(2 + xi - 3   ) 3

                                           1/2
         + 3 ln(2) + 3 ln(3) - 3/2 ln(3 - 3   )

                            1/2                1/2
         - 3/2 ln(2 + xi + 3   ) - 3/2 ln(3 + 3   )

                            1/2
         - 3/2 ln(2 + xi - 3   )

> n:=3:
> r[3][1]:=ln(1/2+xi/2)/(1/2-xi/2);

                                           xi
                                 ln(1/2 + ----)
                                           2
                      r[3][1] := --------------
                                          xi
                                   1/2 - ----
                                          2

> r[3][2]:=ln(1/2+xi/2+(1-xi)*sqrt(3/5)/2)/(1/2-xi/2-(1-xi)*sqrt(3/5)/2)
> ;

                                                    1/2
                                   xi    (1 - xi) 15
                         ln(1/2 + ---- + --------------)
                                   2           10
              r[3][2] := -------------------------------
                                                   1/2
                                  xi    (1 - xi) 15
                           1/2 - ---- - --------------
                                  2           10

> r[3][3]:=ln(1/2+xi/2-(1-xi)*sqrt(3/5)/2)/(1/2-xi/2+(1-xi)*sqrt(3/5)/2)
> ;

                                                    1/2
                                   xi    (1 - xi) 15
                         ln(1/2 + ---- - --------------)
                                   2           10
              r[3][3] := -------------------------------
                                                   1/2
                                  xi    (1 - xi) 15
                           1/2 - ---- + --------------
                                  2           10

> Quadratur[3]:=j*(8/9)*r[3][1]+j*(5/9)*(r[3][2]+r[3][3]);

                      /         xi \           xi
                      |- 1/2 + ----| ln(1/2 + ----)
                      \         2  /           2
  Quadratur[3] := 8/9 ----------------------------- + 5/9
                                      xi
                               1/2 - ----
                                      2

                       /                           1/2
                       |          xi    (1 - xi) 15
                       |ln(1/2 + ---- + --------------)
        /         xi \ |          2           10
        |- 1/2 + ----| |-------------------------------
        \         2  / |                          1/2
                       |         xi    (1 - xi) 15
                       |  1/2 - ---- - --------------
                       \         2           10

                                      1/2 \
                     xi    (1 - xi) 15    |
           ln(1/2 + ---- - --------------)|
                     2           10       |
         + -------------------------------|
                                     1/2  |
                    xi    (1 - xi) 15     |
             1/2 - ---- + --------------  |
                    2           10        /

> quadratur[3]:=simplify(%);

                                                  1/2    1/2
  quadratur[3] := 11/3 ln(2) - 5/18 ln(4 + xi + 15   ) 15

                                         25          1/2
         - 8/9 ln(1 + xi) + 25/9 ln(5) - -- ln(5 - 15   )
                                         18

           25               1/2    25          1/2
         - -- ln(4 + xi + 15   ) - -- ln(5 + 15   )
           18                      18

           25               1/2                       1/2    1/2
         - -- ln(4 + xi - 15   ) + 5/18 ln(4 + xi - 15   ) 15
           18

                         1/2    1/2                 1/2    1/2
         + 5/18 ln(5 + 15   ) 15    - 5/18 ln(5 - 15   ) 15

> plot({0,dilog(xi),Quadratur[1],Quadratur[2],Quadratur[3]},            
>  xi=0..5, scaling=constrained, color=black,axes=boxed); 
> 

> L_zwei:=sqrt((1/4)*Int((Dilogarithmus-Quadratur)^2,xi=1..5));

                     /   5                                 \1/2
                     |  /                                  |
                     | |                              2    |
       L_zwei := 1/2 | |   (Dilogarithmus - Quadratur)  dxi|
                     | |                                   |
                     |/                                    |
                     \  1                                  /

> for i in [1,2,3] do
> L_zwei[i]:=evalf(sqrt((1/4)*int((dilog(xi)-Quadratur[i])^2,xi=1..5)))
> od;
> 

                      L_zwei[1] := 0.08218013370


                     L_zwei[2] := 0.006951010550


                     L_zwei[3] := 0.0006831480505

#    Im betrachteten Bereich  < 1..5 >  ist die Nherung mit einem
# Sttzpunkt zufrieden stellend. Die Nherungen mit  n = 2  und  n = 3 
# GAUSS-Punkten fallen im Rahmen der Strichstrke mit dem Dilogarithmus
# zusammen.
> 
> #   
