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

                     xi
                    /           2
                   |        Pi t
                   |    sin(-----) dt = FresnelS(xi)
                   |          2
                  /
                    0

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

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

> plot({1/2,FresnelS(sqrt(2)),FresnelS(xi)},xi=0..10,numpoints=100,color
> =black);

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

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

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

                    1
                   /           2        2
                  |       Pi xi  (1 + x)
          1/2 xi  |   sin(---------------) dx = FresnelS(xi)
                  |              8
                 /
                   -1

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

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

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

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

  quadratur[2] := 0.500000000000000 xi

                                  2                            2
        (sin(0.0701489345397444 xi ) + sin(0.977048616656859 xi ))

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

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

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

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

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

  quadratur[3] := 0.500000000000000 xi (

                                                  2
        0.888888888888889 sin(0.392699081698724 xi )

                                                      2
         + 0.555555555555556 sin(0.0199517293219168 xi )

                                                    2
         + 0.555555555555556 sin(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]*(sin(Pi*xi^2*(1-X[4])^2/8)+sin(Pi*xi^2*(1+X
> [4])^2/8))+H[4]*(sin(Pi*xi^2*(1-x[4])^2/8)+sin(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         /
        sin(---------------------------------)
                            8

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

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

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

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

  quadratur[4] := 0.500000000000000 xi (

                                                    2
        0.347854845137454 sin(0.00757246507047310 xi )

                                                    2
         + 0.347854845137454 sin(1.36024222019211 xi )

                                                     2
         + 0.652145154862546 sin(0.171069546429280 xi )

                                                     2
         + 0.652145154862546 sin(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;

                                         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

> Quadratur[5]:=(xi/2)*((128/225)*sin(Pi*xi^2/8)+H[5]*(sin(Pi*xi^2*(1-x[
> 5])^2/8)+sin(Pi*xi^2*(1+x[5])^2/8))+h[5]*(sin(Pi*xi^2*(1-X[5])^2/8)+si
> n(Pi*xi^2*(1+X[5])^2/8)));

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

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

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

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

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

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

  quadratur[5] := 0.500000000000000 xi (

                                                  2
        0.568888888888889 sin(0.392699081698724 xi )

                                                      2
         + 0.478628670499366 sin(0.0836490582605305 xi )

                                                     2
         + 0.478628670499366 sin(0.929474672666315 xi )

                                                       2
         + 0.236926885056190 sin(0.00345662422459685 xi )

                                                    2
         + 0.236926885056190 sin(1.42688059764062 xi ))

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

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

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

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

                    L_zwei[1] := 0.592346547492680


                    L_zwei[2] := 0.171406022807502


                   L_zwei[3] := 0.0145516918448778


                   L_zwei[4] := 0.00627424364245075


                  L_zwei[5] := 0.000617824291962020

#    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.
> 
> 
