A115373 -id:A115373 - cp's OEIS frontend

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

Eric W. Weisstein, Jan 21 2006

"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

			2.4048255576957727686...

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.

Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

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

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

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

solve(x=2,3,besselj(0,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(0) \\ Charles R Greathouse IV, Aug 06 2022

A115369 Decimal expansion of first zero of BesselJ(1,z).

Original entry on oeis.org

3, 8, 3, 1, 7, 0, 5, 9, 7, 0, 2, 0, 7, 5, 1, 2, 3, 1, 5, 6, 1, 4, 4, 3, 5, 8, 8, 6, 3, 0, 8, 1, 6, 0, 7, 6, 6, 5, 6, 4, 5, 4, 5, 2, 7, 4, 2, 8, 7, 8, 0, 1, 9, 2, 8, 7, 6, 2, 2, 9, 8, 9, 8, 9, 9, 1, 8, 8, 3, 9, 3, 0, 9, 5, 1, 9, 0, 1, 1, 4, 7, 0, 2, 1, 4, 1, 1, 2, 8, 7, 4, 7, 5, 7, 4, 2, 3, 1, 2, 6, 7, 2, 4, 4, 7

Offset: 1

Eric W. Weisstein, Jan 21 2006

Also the first root of the sinc(2,x) function, that is, the radial component of the 2D Fourier transform of a 2-dimensional unit disc. - Stanislav Sykora, Nov 14 2013

Also the first root of the derivative of BesselJ_0. - Jean-François Alcover, Jul 01 2015

			3.8317059702075123156...

Stanislav Sykora, K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions
Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368, A115370, A115371, A115372, A115373.

Cf. A103365, A103366, A238390, A180874.

BesselJZero[1, 1] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, Feb 06 2013 *)

solve(x=3,4,besselj(1,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(1) \\ Charles R Greathouse IV, Aug 06 2022

A115370 Decimal expansion of first zero of BesselJ(2,z).

Original entry on oeis.org

5, 1, 3, 5, 6, 2, 2, 3, 0, 1, 8, 4, 0, 6, 8, 2, 5, 5, 6, 3, 0, 1, 4, 0, 1, 6, 9, 0, 1, 3, 7, 7, 6, 5, 4, 5, 6, 9, 7, 3, 7, 7, 2, 3, 4, 7, 5, 0, 0, 5, 5, 0, 9, 4, 3, 3, 5, 8, 2, 5, 7, 2, 5, 7, 4, 5, 9, 9, 8, 1, 9, 6, 6, 1, 0, 7, 9, 2, 8, 4, 8, 7, 9, 4, 2, 3, 6, 8, 1, 9, 7, 2, 8, 7, 4, 5, 0, 4, 6, 2, 8, 7, 8, 3, 2

Offset: 1

Eric W. Weisstein, Jan 21 2006

Also the first root of the sinc(4,x) function, that is, the radial component of the 4D Fourier transform of 4-dimensional unit sphere. Also, the solution of 2J_1(x) = x*J_0(x). - Stanislav Sykora, Nov 14 2013

			5.1356223018406825563...

Stanislav Sykora, K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions.
Eric Weisstein's World of Mathematics, Bessel Function Zeros.
Index entries for transcendental numbers

Cf. A115368, A115369, A115371, A115372, A115373.

RealDigits[BesselJZero[2, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=5,6,besselj(2,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(2) \\ Charles R Greathouse IV, Aug 06 2022

A115371 Decimal expansion of first zero of BesselJ(3,z).

Original entry on oeis.org

6, 3, 8, 0, 1, 6, 1, 8, 9, 5, 9, 2, 3, 9, 8, 3, 5, 0, 6, 2, 3, 6, 6, 1, 4, 6, 4, 1, 9, 4, 2, 7, 0, 3, 3, 0, 5, 3, 2, 6, 3, 0, 3, 6, 9, 1, 9, 0, 3, 0, 8, 8, 3, 2, 7, 1, 0, 7, 1, 8, 8, 2, 8, 4, 2, 5, 1, 1, 1, 9, 5, 7, 8, 0, 9, 0, 1, 1, 1, 3, 0, 8, 0, 7, 0, 7, 4, 5, 7, 9, 0, 4, 3, 5, 7, 9, 5, 1, 9, 5, 6, 9, 3, 6, 4

Offset: 1

Eric W. Weisstein, Jan 21 2006

			6.3801618959239835062...

Eric Weisstein's World of Mathematics, Bessel Function Zeros.
Index entries for transcendental numbers

Cf. A115368, A115369, A115370, A115372, A115373.

RealDigits[BesselJZero[3, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=6,7,besselj(3,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(3) \\ Charles R Greathouse IV, Aug 06 2022

A115372 Decimal expansion of first zero of BesselJ(4,z).

Original entry on oeis.org

7, 5, 8, 8, 3, 4, 2, 4, 3, 4, 5, 0, 3, 8, 0, 4, 3, 8, 5, 0, 6, 9, 6, 3, 0, 0, 0, 7, 9, 8, 5, 6, 1, 7, 4, 1, 7, 3, 6, 9, 9, 7, 7, 9, 0, 1, 3, 1, 2, 9, 8, 1, 2, 1, 1, 0, 1, 5, 5, 1, 5, 7, 8, 7, 0, 5, 2, 6, 7, 4, 6, 6, 4, 9, 5, 3, 7, 4, 6, 8, 0, 7, 2, 1, 6, 7, 0, 0, 0, 2, 0, 6, 2, 4, 0, 7, 5, 5, 1, 0, 5, 9, 8, 1, 8

Offset: 1

Eric W. Weisstein, Jan 21 2006

			7.5883424345038043850...

Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368, A115369, A115370, A115371, A115373.

RealDigits[BesselJZero[4,1],10,120][[1]] (* Harvey P. Dale, Jan 16 2012 *)

solve(x=7,8,besselj(4,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(4) \\ Charles R Greathouse IV, Aug 06 2022

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

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A115369 Decimal expansion of first zero of BesselJ(1,z).

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A115370 Decimal expansion of first zero of BesselJ(2,z).

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A115371 Decimal expansion of first zero of BesselJ(3,z).

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A115372 Decimal expansion of first zero of BesselJ(4,z).

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