A217481 Decimal expansion of sqrt(2*Pi)/4.

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, 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, 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

Offset: 0

Author

R. J. Mathar, Oct 04 2012

Keywords

• cons
• nonn

Comments

Equals Integral_{x>=0} sin(x^2) dx.

The generalizations are Integral_{x>=0} exp(i*x^n) dx =

0.6266570686577501... + i*0.6266570686577501... for n=2,

0.7733429420779898... + i*0.4464897557846246... for n=3,

0.8374066967690864... + i*0.3468652110238094... for n=4,

0.8732303655178185... + i*0.2837297451053993... for n=5,

and

Gamma(1/n)*i^(1/n)/n in general, where i is the imaginary unit. - R. J. Mathar, Nov 14 2012

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

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

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

Examples

			equals 0.62665706865775012560394132120276131... = A019727 / 4 = sqrt(A019675).
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2000
• Robert Ferréol, Cornu spiral, Mathcurve.
• I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).
• Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.4.7)
• Wikipedia, Fresnel Integral
• Index entries for transcendental numbers

Crossrefs

Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, this sequence, A019706, A143149, A019710.

Programs

Magma

Sqrt(2*Pi(RealField(100)))/4; // G. C. Greubel, Sep 30 2018

Maple

evalf(sqrt(2*Pi))/4 ;

Mathematica

First@ RealDigits[N[Sqrt[2 Pi]/4, 105]] (* Michael De Vlieger, Sep 24 2018 *)

Maxima

fpprec : 100; ev(bfloat(sqrt(2*%pi)))/4; /* Martin Ettl, Oct 04 2012 */

PARI

sqrt(2*Pi)/4 \\ Altug Alkan, Sep 08 2018

Sage

((sqrt(2*pi))/4).n(digits=100) # Jani Melik, Oct 05 2012

Formula

From A.H.M. Smeets, Sep 22 2018: (Start)

Equals Integral_{x >= 0} sin(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik].

Equals Integral_{x >= 0} cos(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik]. (End)

From Bernard Schott, Mar 02 2020: (Start)

Equals Integral_{x >= 0} cos(x^2) dx or Integral_{x >= 0} sin(x^2) dx.

Equals sqrt(Pi/8) or (1/2)*sqrt(Pi/2). (End)