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A217481
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Decimal expansion of sqrt(2*Pi)/4.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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)
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EXAMPLE
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equals 0.62665706865775012560394132120276131... = A019727 / 4 = sqrt(A019675).
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MAPLE
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evalf(sqrt(2*Pi))/4 ;
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MATHEMATICA
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PROG
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(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
(Magma) Sqrt(2*Pi(RealField(100)))/4; // G. C. Greubel, Sep 30 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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