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%I #33 Aug 06 2022 21:58:58
%S 2,4,0,4,8,2,5,5,5,7,6,9,5,7,7,2,7,6,8,6,2,1,6,3,1,8,7,9,3,2,6,4,5,4,
%T 6,4,3,1,2,4,2,4,4,9,0,9,1,4,5,9,6,7,1,3,5,7,0,6,9,9,9,0,9,0,5,9,6,7,
%U 6,5,8,3,8,6,7,7,1,9,4,0,2,9,2,0,4,4,3,6,3,4,3,7,6,0,1,4,5,2,5,4,7,8,6,8,9
%N Decimal expansion of first zero of the Bessel function J_0(z).
%C "This [constant] arises from the study of a vibrating, homogeneous membrane that is uniformly stretched across the unit disk. [Its square] is the principal frequency of the sound one hears when a kettledrum is struck." - Quoted from the book by Steven R. Finch.
%C Siegel proves (the Main Theorem) that J_0(z) is transcendental if z is algebraic and nonzero, but since in our case J_0(z) = 0 is not transcendental it follows that z cannot be algebraic. - _Charles R Greathouse IV_, Oct 20 2020
%D Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 7.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 221.
%D C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1. Translated as "On some applications
%D of Diophantine approximations" by Clemens Fuchs.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BesselFunctionZeros.html">Bessel Function Zeros</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e 2.4048255576957727686...
%t RealDigits[BesselJZero[0, 1], 10, 120][[1]] (* _Alonso del Arte_, May 06 2011 *)
%o (PARI) solve(x=2,3,besselj(0,x)) \\ _Charles R Greathouse IV_, Feb 19 2014
%o (PARI) besseljzero(0) \\ _Charles R Greathouse IV_, Aug 06 2022
%Y Cf. A115369, A115370, A115371, A115372, A115373.
%Y Cf. A000275, A002190, A188489, A181167, A181168.
%K nonn,cons
%O 1,1
%A _Eric W. Weisstein_, Jan 21 2006
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