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

0,1

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

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

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)

Example

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

Maple

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

Mathematica

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

Prog

(Maxima) fpprec : 100; ev(bfloat(sqrt(2*%pi)))/4; /* Martin Ettl, Oct 04 2012 */
(Sage) ((sqrt(2*pi))/4).n(digits=100) # Jani Melik, Oct 05 2012
(PARI) sqrt(2*Pi)/4 \\ Altug Alkan, Sep 08 2018
(Magma) Sqrt(2*Pi(RealField(100)))/4; // G. C. Greubel, Sep 30 2018

Crossrefs

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

Sequence in context: A011005 A292178 A281961 * A318300 A278146 A221716

Adjacent sequences: A217478 A217479 A217480 * A217482 A217483 A217484

Keyword

cons,nonn

Author

R. J. Mathar, Oct 04 2012

Status

approved