# FRESNELsches Integral C(xi):
> Int(cos(Pi*t^2/2),t=0..xi)=int(cos(Pi*t^2/2),t=0..xi);

                     xi
                    /           2
                   |        Pi t
                   |    cos(-----) dt = FresnelC(xi)
                   |          2
                  /
                    0

> Limit(FresnelC(xi),xi=infinity): % = value (%);

                       lim        FresnelC(xi) = 1/2
                  xi -> infinity

# Einhllende des FRESNELschen Integrals C(xi):
> Y(xi):=1/2+A*exp(-B*xi)+C*exp(-D_*xi);
> A:=0.45793743891023; B:=0.88290504747386;
> C:=0.10121459861159;  D_:=0.11378765780382;

             Y(xi) := 1/2 + A exp(-B xi) + C exp(-D_ xi)


                        A := 0.45793743891023


                        B := 0.88290504747386


                        C := 0.10121459861159


                        D_ := 0.11378765780382

> y(xi):=1/2+a*exp(-b*xi)+c*exp(-d*xi);
> a:=-0.40399067807815; b:=0.83757109631419;
> c:=-0.10355947371466; d:=0.11915836569374;

              y(xi) := 1/2 + a exp(-b xi) + c exp(-d xi)


                        a := -0.40399067807815


                        b := 0.83757109631419


                        c := -0.10355947371466


                        d := 0.11915836569374

> Y(xi):=1/2+A*exp(-B*xi)+C*exp(-D_*xi):
> y(xi):=1/2+a*exp(-b*xi)+c*exp(-d*xi):                  
> plot({1/2,Y(xi),y(xi),FresnelC(xi)}, xi=0..10,numpoints=100,
> color=black);

> Y(infinity):=evalf(subs(xi=infinity,Y(xi))); 

                     Y(infinity) := 0.5000000000

> y(infinity):=evalf(subs(xi=infinity,y(xi)));

                     y(infinity) := 0.5000000000

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

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

> (xi/2)*Int(cos((Pi/2)*t^2),x=-1..1)=
> radsimp(int((xi/2)*cos((Pi/2)*t^2),x=-1..1));

                    1
                   /           2        2
                  |       Pi xi  (1 + x)
          1/2 xi  |   cos(---------------) dx = FresnelC(xi)
                  |              8
                 /
                   -1

> Digits:=15:                                                           
>       n:=1:                                           
> Quadratur[1]:=xi*cos(Pi*xi^2/8);

                                                2
                                           Pi xi
                    Quadratur[1] := xi cos(------)
                                             8

> n:=2:
> Quadratur[2]:=(xi/2)*(cos(Pi*xi^2*(1-1/sqrt(3))^2/8)+cos(Pi*xi^2*(1+1/
> sqrt(3))^2/8));

  Quadratur[2] :=

               /           /     1/2\2               /     1/2\2 \
               |         2 |    3   |              2 |    3   |  |
               |    Pi xi  |1 - ----|         Pi xi  |1 + ----|  |
               |           \     3  /                \     3  /  |
        1/2 xi |cos(------------------) + cos(------------------)|
               \            8                         8          /

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

  quadratur[2] := 0.500000000000000 xi

                                  2                            2
        (cos(0.0701489345397444 xi ) + cos(0.977048616656859 xi ))

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

                         /
                         |
                         |             2
                         |        Pi xi
  Quadratur[3] := 1/2 xi |8/9 cos(------)
                         \          8

                          /      1/2\2
                        2 |    15   |
                   Pi xi  |1 - -----|
                          \      5  /
         + 5/9 cos(-------------------)
                            8

                          /      1/2\2 \
                        2 |    15   |  |
                   Pi xi  |1 + -----|  |
                          \      5  /  |
         + 5/9 cos(-------------------)|
                            8          /

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

  quadratur[3] := 0.500000000000000 xi (

                                                  2
        0.888888888888889 cos(0.392699081698724 xi )

                                                      2
         + 0.555555555555556 cos(0.0199517293219168 xi )

                                                    2
         + 0.555555555555556 cos(1.23668533211399 xi ))

> n:=4:
> X[4]:=sqrt(525+70*sqrt(30))/35;  x[4]:=sqrt(525-70*sqrt(30))/35; 
> H[4]:=1/2+sqrt(30)/36;           h[4]:=1/2-sqrt(30)/36;

                                         1/2 1/2
                             (525 + 70 30   )
                     X[4] := -------------------
                                     35


                                         1/2 1/2
                             (525 - 70 30   )
                     x[4] := -------------------
                                     35


                                         1/2
                                       30
                         H[4] := 1/2 + -----
                                        36


                                         1/2
                                       30
                         h[4] := 1/2 - -----
                                        36

> Quadratur[4]:=(xi/2)*(h[4]*(cos(Pi*xi^2*(1-X[4])^2/8)+cos(Pi*xi^2*(1+X
> [4])^2/8))+H[4]*(cos(Pi*xi^2*(1-x[4])^2/8)+cos(Pi*xi^2*(1+x[4])^2/8)))
> ;              

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

                   /                1/2 1/2\2
                 2 |    (525 + 70 30   )   |
            Pi xi  |1 - -------------------|
                   \            35         /
        cos(---------------------------------)
                            8

                      /                1/2 1/2\2 \                 /
                    2 |    (525 + 70 30   )   |  |                 |
               Pi xi  |1 + -------------------|  |   /        1/2\ |
                      \            35         /  |   |      30   | |
         + cos(---------------------------------)| + |1/2 + -----| |
                               8                 /   \       36  / \

                   /                1/2 1/2\2
                 2 |    (525 - 70 30   )   |
            Pi xi  |1 - -------------------|
                   \            35         /
        cos(---------------------------------)
                            8

                      /                1/2 1/2\2 \\
                    2 |    (525 - 70 30   )   |  ||
               Pi xi  |1 + -------------------|  ||
                      \            35         /  ||
         + cos(---------------------------------)||
                               8                 //

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

  quadratur[4] := 0.500000000000000 xi (

                                                    2
        0.347854845137454 cos(0.00757246507047310 xi )

                                                    2
         + 0.347854845137454 cos(1.36024222019211 xi )

                                                     2
         + 0.652145154862546 cos(0.171069546429280 xi )

                                                     2
         + 0.652145154862546 cos(0.705110520872271 xi ))

> n:=5:
> X[5]:=sqrt(245+14*sqrt(70))/21;  x[5]:=sqrt(245-14*sqrt(70))/21;
> H[5]:=161/450+13*sqrt(70)/900;  h[5]:=161/450-13*sqrt(70)/900;
> Quadratur[5]:=(xi/2)*((128/225)*cos(Pi*xi^2/8)+H[5]*(cos(Pi*xi^2*(1-x[
> 5])^2/8)+cos(Pi*xi^2*(1+x[5])^2/8))+h[5]*(cos(Pi*xi^2*(1-X[5])^2/8)+co
> s(Pi*xi^2*(1+X[5])^2/8)));

                                         1/2 1/2
                             (245 + 14 70   )
                     X[5] := -------------------
                                     21


                                         1/2 1/2
                             (245 - 14 70   )
                     x[5] := -------------------
                                     21


                                           1/2
                                161   13 70
                        H[5] := --- + --------
                                450     900


                                           1/2
                                161   13 70
                        h[5] := --- - --------
                                450     900


                         /                                   /
                         |                                   |
                         |             2    /           1/2\ |
                         |128     Pi xi     |161   13 70   | |
  Quadratur[5] := 1/2 xi |--- cos(------) + |--- + --------| |
                         \225       8       \450     900   / \

                   /                1/2 1/2\2
                 2 |    (245 - 14 70   )   |
            Pi xi  |1 - -------------------|
                   \            21         /
        cos(---------------------------------)
                            8

                      /                1/2 1/2\2 \
                    2 |    (245 - 14 70   )   |  |
               Pi xi  |1 + -------------------|  |   /           1/2\
                      \            21         /  |   |161   13 70   |
         + cos(---------------------------------)| + |--- - --------|
                               8                 /   \450     900   /

        /           /                1/2 1/2\2
        |         2 |    (245 + 14 70   )   |
        |    Pi xi  |1 - -------------------|
        |           \            21         /
        |cos(---------------------------------)
        \                    8

                      /                1/2 1/2\2 \\
                    2 |    (245 + 14 70   )   |  ||
               Pi xi  |1 + -------------------|  ||
                      \            21         /  ||
         + cos(---------------------------------)||
                               8                 //

> quadratur[5]:=evalf(%);

  quadratur[5] := 0.500000000000000 xi (

                                                  2
        0.568888888888889 cos(0.392699081698724 xi )

                                                      2
         + 0.478628670499366 cos(0.0836490582605305 xi )

                                                     2
         + 0.478628670499366 cos(0.929474672666315 xi )

                                                       2
         + 0.236926885056190 cos(0.00345662422459685 xi )

                                                    2
         + 0.236926885056190 cos(1.42688059764062 xi ))

> plot({1,FresnelC(2),FresnelC(infinity),FresnelC(xi),Quadratur[1],Quadr
> atur[2],                                                              
>                                                                       
>         Quadratur[3],                     
> Quadratur[4],Quadratur[5]},xi=0..2,scaling=constrained,color=black);

> L_zwei:=sqrt((1/2)*Int((FresnelC(xi)-Quadratur)^2,xi=0..2));

                        /   2                                \1/2
                        |  /                                 |
                    1/2 | |                             2    |
     L_zwei := 1/2 2    | |   (FresnelC(xi) - Quadratur)  dxi|
                        | |                                  |
                        |/                                   |
                        \  0                                 /

> for i in [1,2,3,4,5] do                      
> L_zwei[i]:=evalf(sqrt((1/2)*int((FresnelC(xi)-Quadratur[i])^2,xi=0..2)
> ))       od;

                    L_zwei[1] := 0.256683296889972


                    L_zwei[2] := 0.151607163842766


                   L_zwei[3] := 0.0456292912497340


                   L_zwei[4] := 0.00336360499195284


                  L_zwei[5] := 0.000523687823188925

#    Im Integrationsinterval  [0,2]  liefert die GAUSS-Quadratur mit   n
# = 4   und  n = 5  Sttzstellen sehr gute Nherungen. Im weiteren
# Verlauf  [2, >2]  ergeben sich groe Abweichungen.
> 
> 
