OFFSET
0,1
COMMENTS
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
REFERENCES
C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 210.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).
Michael Penn, An interesting approach to the Gaussian integral, YouTube video, 2021.
Eric Weisstein's World of Mathematics, Fractional Calculus.
FORMULA
Equals (1/2)! = Gamma(3/2). - Benoit Cloitre, Apr 24 2003
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
Equals sqrt(A003881). - Michel Marcus, Aug 31 2014
From A.H.M. Smeets, Sep 22 2018: (Start)
Equals Integral_{x >= 0} sin(2x)/sqrt(x) dx [Gradshteyn and Ryzhik].
Equals Integral_{x >= 0} cos(2x)/sqrt(x) dx [Gradshteyn and Ryzhik]. (End)
Equals Integral_{x=0..oo} sin(x^2)^2/x^2 dx. - Amiram Eldar, Aug 21 2020
EXAMPLE
sqrt(Pi)/2 = 0.886226925452758013649...
MAPLE
evalf(sqrt(Pi)/2, 120); # Muniru A Asiru, Sep 22 2018
MATHEMATICA
RealDigits[Sqrt[Pi]/2, 10, 100][[1]] (* Alonso del Arte, Aug 15 2012 *)
PROG
(PARI) gammah(1) \\ Michel Marcus, Feb 11 2016
(PARI) sqrt(Pi)/2 \\ Michel Marcus, Feb 11 2016
(PARI) intnum(x=0, [oo, -2*I], sin(2*x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018
(PARI) intnum(x=[0, -1/2], [oo, 2*I], cos(2*x)/sqrt(x)) \\ Gheorghe Coserea, Sep 23 2018
(PARI) intnum(x=1, [oo, 1], exp(-(x-1/x)^2)*(1 + 1/x^2)) \\ Gheorghe Coserea, Sep 24 2018
(Magma) pi:=Sqrt(Pi(RealField(110)))/ 2; Reverse(Intseq(Floor(10^110*pi))); // Vincenzo Librandi, Feb 11 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved