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A217481 Decimal expansion of sqrt(2*Pi)/4. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
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

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..2000

I. S. Gradsteyn, I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).

Wikipedia, Fresnel Integral

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)

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

CROSSREFS

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

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)