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%I #21 Mar 06 2015 10:53:44
%S 1,6,1,6,3,3,9,9,4,8,3,1,0,7,0,3,1,7,8,1,1,9,1,3,9,7,5,3,6,8,3,8,9,6,
%T 3,0,9,7,4,3,1,2,1,0,9,7,2,1,5,4,6,1,0,2,3,5,8,1,1,4,3,6,6,2,1,7,7,2,
%U 2,6,4,3,7,0,7,7
%N Decimal expansion of the abscissa of the half width of the Airy function.
%C In optics, the Airy function is the amplitude pattern of light shining through a circular hole, which gives (in the Fraunhofer limit of diffraction theory) an amplitude proportional to J_1(z)/z, where J_1 is the Bessel function of order 1, and where z is the radial coordinate. The Airy disk is the intensity, the square of the amplitude, proportional to I=[J_1(z)/z)]^2, with a first zero a A115369. The peak is at I(0)=1/4, so the half width is defined by I(zhalf)=1/8, which gives zhalf = 1.6163399483.., defining the sequence of digits.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Airy_disk">Airy disk</a>
%t z /. FindRoot[ BesselJ[1, z]^2/z^2 == 1/8 , {z, 1}, WorkingPrecision -> 76] // RealDigits // First (* _Jean-François Alcover_, Feb 21 2013 *)
%o (PARI) solve(x=1,2,8*besselj(1,x)^2-x^2) \\ _Charles R Greathouse IV_, Feb 19 2014
%Y Cf. A245461.
%K nonn,cons
%O 1,2
%A _R. J. Mathar_, Feb 21 2013
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