|
%I #96 Sep 08 2022 08:46:04
%S 6,2,6,6,5,7,0,6,8,6,5,7,7,5,0,1,2,5,6,0,3,9,4,1,3,2,1,2,0,2,7,6,1,3,
%T 1,3,2,5,1,7,4,6,6,8,5,1,5,2,4,8,4,5,7,9,1,5,7,4,8,0,8,9,4,0,8,5,5,7,
%U 3,4,1,3,6,5,1,9,6,0,4,9,3,7,3,6,6,4,8,9,5,9,5,9,4,5,1,4,3,1,6,5,2,9,0,0,2
%N Decimal expansion of sqrt(2*Pi)/4.
%C Equals Integral_{x>=0} sin(x^2) dx.
%C The generalizations are Integral_{x>=0} exp(i*x^n) dx =
%C 0.6266570686577501... + i*0.6266570686577501... for n=2,
%C 0.7733429420779898... + i*0.4464897557846246... for n=3,
%C 0.8374066967690864... + i*0.3468652110238094... for n=4,
%C 0.8732303655178185... + i*0.2837297451053993... for n=5,
%C and
%C Gamma(1/n)*i^(1/n)/n in general, where i is the imaginary unit. - _R. J. Mathar_, Nov 14 2012
%C Mean of cycle length (and of tail length) in Pollard rho method for factoring n is sqrt(2*Pi)/4*sqrt(n). - _Jean-François Alcover_, May 27 2013
%C If m = (1/2) * sqrt(Pi/2), then the coordinates of the two asymptotic points of the Cornu spiral (also called clothoide) and whose Cartesian parametrization is: x = a * Integral_{0..t} cos(u^2) du and y = a * Integral_{0..t} sin(u^2) du are (a*m, a*m) and (-a*m, -a*m) (see the curve at the MathCurve link). - _Bernard Schott_, Mar 02 2020
%C Equals the limit as x approaches infinity of the Fresnel integrals Integral_{0..x} sin(t^2) dt and Integral_{0..x} cos(t^2) dt. - _Bernard Schott_, Mar 05 2020
%H Muniru A Asiru, <a href="/A217481/b217481.txt">Table of n, a(n) for n = 0..2000</a>
%H Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/cornu/cornu.shtml">Cornu spiral</a>, Mathcurve.
%H I. S. Gradsteyn and I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a>, (1980), page 420 (formulas 3.757.1, 3.757.2).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fresnel_integral">Fresnel Integral</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F From _A.H.M. Smeets_, Sep 22 2018: (Start)
%F Equals Integral_{x >= 0} sin(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik].
%F Equals Integral_{x >= 0} cos(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik]. (End)
%F From _Bernard Schott_, Mar 02 2020: (Start)
%F Equals Integral_{x >= 0} cos(x^2) dx or Integral_{x >= 0} sin(x^2) dx.
%F Equals sqrt(Pi/8) or (1/2)*sqrt(Pi/2). (End)
%e equals 0.62665706865775012560394132120276131... = A019727 / 4 = sqrt(A019675).
%p evalf(sqrt(2*Pi))/4 ;
%t First@ RealDigits[N[Sqrt[2 Pi]/4, 105]] (* _Michael De Vlieger_, Sep 24 2018 *)
%o (Maxima) fpprec : 100; ev(bfloat(sqrt(2*%pi)))/4; /* _Martin Ettl_, Oct 04 2012 */
%o (Sage) ((sqrt(2*pi))/4).n(digits=100) # _Jani Melik_, Oct 05 2012
%o (PARI) sqrt(2*Pi)/4 \\ _Altug Alkan_, Sep 08 2018
%o (Magma) Sqrt(2*Pi(RealField(100)))/4; // _G. C. Greubel_, Sep 30 2018
%Y Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, this sequence, A019706, A143149, A019710.
%K cons,nonn
%O 0,1
%A _R. J. Mathar_, Oct 04 2012
|
