A221210 Decimal expansion of the abscissa of the half width of the Airy function.

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

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 (* Jean-François Alcover, Feb 21 2013 *)

Prog

(PARI) solve(x=1, 2, 8*besselj(1, x)^2-x^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