

A221210


Decimal expansion of the abscissa of the half width of the Airy function.


1



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, 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, 2, 6, 4, 3, 7, 0, 7, 7
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OFFSET

1,2


COMMENTS

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.


LINKS

Table of n, a(n) for n=1..76.
Wikipedia, Airy disk


MATHEMATICA

z /. FindRoot[ BesselJ[1, z]^2/z^2 == 1/8 , {z, 1}, WorkingPrecision > 76] // RealDigits // First (* JeanFrançois Alcover, Feb 21 2013 *)


PROG

(PARI) solve(x=1, 2, 8*besselj(1, x)^2x^2) \\ Charles R Greathouse IV, Feb 19 2014


CROSSREFS

Cf. A245461.
Sequence in context: A078300 A176398 A318478 * A010492 A276515 A144544
Adjacent sequences: A221207 A221208 A221209 * A221211 A221212 A221213


KEYWORD

nonn,cons


AUTHOR

R. J. Mathar, Feb 21 2013


STATUS

approved



