A115368 Decimal expansion of first zero of the Bessel function J_0(z).

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, 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, 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

Offset

1,1

Comments

"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.

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

References

Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 7.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 221.

C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1. Translated as "On some applications

of Diophantine approximations" by Clemens Fuchs.

Links

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Bessel Function Zeros

Index entries for transcendental numbers

Example

2.4048255576957727686...

Mathematica

RealDigits[BesselJZero[0, 1], 10, 120][[1]] (* Alonso del Arte, May 06 2011 *)

Prog

(PARI) solve(x=2, 3, besselj(0, x)) \\ Charles R Greathouse IV, Feb 19 2014
(PARI) besseljzero(0) \\ Charles R Greathouse IV, Aug 06 2022

Crossrefs

Cf. A115369, A115370, A115371, A115372, A115373.

Cf. A000275, A002190, A188489, A181167, A181168.

Sequence in context: A327898 A140875 A364315 * A086932 A331762 A221255

Adjacent sequences: A115365 A115366 A115367 * A115369 A115370 A115371

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 21 2006

Status

approved