# Integrallogarithmus li(xi):  
> Int(1/ln(t),t=0..xi)=int(1/ln(t),t=0..xi);

                      xi
                     /
                    |      1
                    |    ----- dt = -Ei(1, -ln(xi))
                    |    ln(t)
                   /
                     0

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

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

> (xi/2)*Int(1/ln(xi*(1+x)/2),x=-1..1)=           
> (xi/2)*simplify(int(1/ln(xi*(1+x)/2),x=-1..1));

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

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

> Digits:=5:
> n:=1:
> Quadratur[1]:=xi/ln(xi/2);

                                          xi
                       Quadratur[1] := --------
                                           xi
                                       ln(----)
                                           2

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

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

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

                             0.50000 xi       0.50000 xi
           quadratur[2] := -------------- + --------------
                           ln(0.21132 xi)   ln(0.78870 xi)

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

                         xi                   xi
  Quadratur[3] := 4/9 -------- + 5/18 ------------------
                          xi                /      1/2\
                      ln(----)              |    15   |
                          2              xi |1 - -----|
                                            \      5  /
                                      ln(--------------)
                                               2

                        xi
         + 5/18 ------------------
                      /      1/2\
                      |    15   |
                   xi |1 + -----|
                      \      5  /
                ln(--------------)
                         2

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

                     0.44444 xi       0.27778 xi       0.27778 xi
   quadratur[3] := -------------- + -------------- + --------------
                   ln(0.50000 xi)   ln(0.11270 xi)   ln(0.88730 xi)

> n:=4:
> g:=sqrt(525-70*sqrt(30))/35;                        
> G:=sqrt(525+70*sqrt(30))/35;             
> H:=(490+49*sqrt(30))/(900+48*sqrt(30));                            
> H:=factor(H);                            
> h:=(490-49*sqrt(30))/(900-48*sqrt(30));                           
> h:=factor(h);

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


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


                                         1/2
                              490 + 49 30
                         H := --------------
                                         1/2
                              900 + 48 30


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


                                         1/2
                              490 - 49 30
                         h := --------------
                                         1/2
                              900 - 48 30


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

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

                              /        1/2\
                              |      30   |
                              |1/2 - -----| xi
                              \       36  /
  Quadratur[4] := 1/2 --------------------------------
                            /                1/2 1/2\
                            |    (525 + 70 30   )   |
                         xi |1 - -------------------|
                            \            35         /
                      ln(----------------------------)
                                      2

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

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

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

# 
> quadratur[4]:=evalf(%);

                    0.17392 xi        0.17392 xi       0.32608 xi
  quadratur[4] := --------------- + -------------- + --------------
                  ln(0.069435 xi)   ln(0.93055 xi)   ln(0.33000 xi)

             0.32608 xi
         + --------------
           ln(0.67000 xi)

> plot({-5,-Ei(1,-ln(xi)),Quadratur[1],Quadratur[2],Quadratur[3],Quadrat
> ur[4]}, xi=0..1,-5..0,color=black);

> L_zwei:=sqrt((5/3)*Int((Integrallogarithmus-Quadratur)^2,xi=1/5..4/5))
> ;

  L_zwei :=

                  /   4/5                                       \1/2
                  |  /                                          |
              1/2 | |                                      2    |
        1/3 15    | |     (Integrallogarithmus - Quadratur)  dxi|
                  | |                                           |
                  |/                                            |
                  \  1/5                                        /

> Digits:=10:
> for i in [1,2,3,4] do                 
> L_zwei[i]:=evalf(sqrt((5/3)*int((-Ei(1,-ln(xi))-Quadratur[i])^2,xi=1/5
> ..4/5)))           od;
> 

                      L_zwei[1] := 0.08235827686


                     L_zwei[2] := 0.009514531952


                     L_zwei[3] := 0.001089584911

Warning, computation interrupted

#    Im Bereich   <  1/5 ... 4/5 >  weichen die Nherungen nur
# geringfgig vom Integrallogarithmus ab.                  
> #                                                                     
#      
> Grenzwert:=Limit(-Ei(1,-ln(xi)),xi=1);
> Grenzwert_genhert:=-Ei(1,-ln(0.9999999999));

                  Grenzwert_genhert := -22.44863527

