A019704 - OEIS

%I #104 Oct 21 2024 17:29:26

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

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

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

%N Decimal expansion of sqrt(Pi)/2.

%C Also half integral of f(x)=sqrt(x) with 0<=x<=1 and half derivative of the same f(x). See Fractional Calculus link. - _Andrea Pinos_, Jul 15 2023

%D C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 210.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.

%H Ivan Panchenko, <a href="/A019704/b019704.txt">Table of n, a(n) for n = 0..1000</a>

%H I. S. Gradsteyn and I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a>, (1980), page 420 (formulas 3.757.1, 3.757.2).

%H Michael Penn, <a href="https://www.youtube.com/watch?v=_7TlAzfs_9Q">An interesting approach to the Gaussian integral</a>, YouTube video, 2021.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FractionalCalculus.html">Fractional Calculus</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%H <a href="/index/Ga#gamma_function">Index to sequences related to the Gamma function</a>.

%F Equals (1/2)! = Gamma(3/2). - _Benoit Cloitre_, Apr 24 2003

%F Equals Integral_{x=0..oo} exp(-x^2) dx = Integral_{x=0..oo} exp(-(x - 1/x)^2) dx = Integral_{x=0..1} sqrt(log(1/x)) dx. - _Jean-François Alcover_, Mar 28 2013

%F Equals sqrt(A003881). - _Michel Marcus_, Aug 31 2014

%F From _A.H.M. Smeets_, Sep 22 2018: (Start)

%F Equals Integral_{x >= 0} sin(2x)/sqrt(x) dx [Gradshteyn and Ryzhik].

%F Equals Integral_{x >= 0} cos(2x)/sqrt(x) dx [Gradshteyn and Ryzhik]. (End)

%F Equals Integral_{x=0..oo} sin(x^2)^2/x^2 dx. - _Amiram Eldar_, Aug 21 2020

%e sqrt(Pi)/2 = 0.886226925452758013649...

%p evalf(sqrt(Pi)/2,120); # _Muniru A Asiru_, Sep 22 2018

%t RealDigits[Sqrt[Pi]/2, 10, 100][[1]] (* _Alonso del Arte_, Aug 15 2012 *)

%o (PARI) gammah(1) \\ _Michel Marcus_, Feb 11 2016

%o (PARI) sqrt(Pi)/2 \\ _Michel Marcus_, Feb 11 2016

%o (PARI) intnum(x=0, [oo, -2*I], sin(2*x)/sqrt(x)) \\ _Gheorghe Coserea_, Sep 23 2018

%o (PARI) intnum(x=[0,-1/2], [oo, 2*I], cos(2*x)/sqrt(x)) \\ _Gheorghe Coserea_, Sep 23 2018

%o (PARI) intnum(x=1, [oo, 1], exp(-(x-1/x)^2)*(1 + 1/x^2)) \\ _Gheorghe Coserea_, Sep 24 2018

%o (Magma) pi:=Sqrt(Pi(RealField(110)))/ 2; Reverse(Intseq(Floor(10^110*pi))); // _Vincenzo Librandi_, Feb 11 2016

%Y Cf. A003881.

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_