A019710 -id:A019710 - cp's OEIS frontend

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

R. J. Mathar, Oct 04 2012

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

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

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

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

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

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

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

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

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

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

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)

A143149 Decimal expansion of 5sqrt(2Pi)/4.

Original entry on oeis.org

3, 1, 3, 3, 2, 8, 5, 3, 4, 3, 2, 8, 8, 7, 5, 0, 6, 2, 8, 0, 1, 9, 7, 0, 6, 6, 0, 6, 0, 1, 3, 8, 0, 6, 5, 6, 6, 2, 5, 8, 7, 3, 3, 4, 2, 5, 7, 6, 2, 4, 2, 2, 8, 9, 5, 7, 8, 7, 4, 0, 4, 4, 7, 0, 4, 2, 7, 8, 6, 7, 0, 6, 8, 2, 5, 9, 8, 0, 2, 4, 6, 8, 6, 8, 3, 2, 4, 4, 7, 9, 7, 9, 7, 2, 5, 7, 1, 5, 8, 2, 6, 4, 5

Offset: 1

Jonathan Vos Post, Jul 27 2008

Upper bound using Shannon entropy arising in randomly-projected hypercubes.

			3.13328534328875...

David L. Donoho and Jared Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete & computational geometry, Vol. 43 (2010), pp. 522-541, see p. 533; arXiv preprint, arXiv:0807.3590 [math.MG], 2008, see p. 10.
Wikipedia, Fresnel Integral.
Index entries for transcendental numbers.

Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, A217481, A019706, this sequence, A019710.

Cf. A143148 (lower bound).

RealDigits[5*Sqrt[2*Pi]/4, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

5*sqrt(2*Pi)/4 \\ Michel Marcus, Mar 06 2020

Equals 10*Integral_{x>=0} x*sin(x^4) dx or 10*Integral_{x>=0} x*cos(x^4) dx (Fresnel integrals).

Edited and a(100) corrected by Georg Fischer, Jul 16 2021

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

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A143149 Decimal expansion of 5sqrt(2Pi)/4.

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

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A143149 Decimal expansion of 5*sqrt(2*Pi)/4.

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A143149 Decimal expansion of 5sqrt(2Pi)/4.