A115369 -id:A115369 - cp's OEIS frontend

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

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

A115373 Decimal expansion of first zero of BesselJ(5,z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

			8.7714838159599540191...

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

Cf. A115368, A115369, A115370, A115371, A115372.

RealDigits[BesselJZero[5, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=8,9,besselj(5,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(5) \\ Charles R Greathouse IV, Aug 23 2022

A259616 Decimal expansion of J'_1(1), the first root of the derivative of the Bessel function J_1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 01 2015

Also root of the equation J_0(x) = J_2(x). - Vaclav Kotesovec, Jul 01 2015

			1.8411837813406593026436295136444433224361270390968192643504677429242292...

Eric Weisstein's MathWorld, Bessel Function Zeros

Cf. A115369 J'_0(1), A259617 J'_2(1), A259618 J'_3(1), A259619 J'_4(1), A259620 J'_5(1).

FindRoot[D[BesselJ[1, x], x] == 0, {x, 2}, WorkingPrecision -> 105] // Last // Last // RealDigits // First

A259617 Decimal expansion of J'_2(1), the first root of the derivative of the Bessel function J_2.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 01 2015

			3.054236928227140322755932091148560897641496760529945919816437566658545...

Eric Weisstein's MathWorld, Bessel Function Zeros

Cf. A115369 J'_0(1), A259616 J'_1(1), A259618 J'_3(1), A259619 J'_4(1), A259620 J'_5(1).

FindRoot[D[BesselJ[2, x], x] == 0, {x, 3}, WorkingPrecision -> 105] // Last // Last // RealDigits // First

A259618 Decimal expansion of J'_3(1), the first root of the derivative of the Bessel function J_3.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 01 2015

			4.2011889412105284961878552974567121879446032135899833521760017910209584...

Eric Weisstein's MathWorld, Bessel Function Zeros

Cf. A115369 J'_0(1), A259616 J'_1(1), A259617 J'_2(1), A259619 J'_4(1), A259620 J'_5(1).

FindRoot[D[BesselJ[3, x], x] == 0, {x, 4}, WorkingPrecision -> 104] // Last // Last // RealDigits // First

A259619 Decimal expansion of J'_4(1), the first root of the derivative of the Bessel function J_4.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 01 2015

			5.317553126083994350363355558188919214766698286220102866220429337604059...

Eric Weisstein's MathWorld, Bessel Function Zeros

Cf. A115369 J'_0(1), A259616 J'_1(1), A259617 J'_2(1), A259618 J'_3(1), A259620 J'_5(1).

FindRoot[D[BesselJ[4, x], x] == 0, {x, 5}, WorkingPrecision -> 104] // Last // Last // RealDigits // First

A259620 Decimal expansion of J'_5(1), the first root of the derivative of the Bessel function J_5.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 01 2015

			6.41561637570024028283981471908792403810900056520772077867494762732288...

Eric Weisstein's MathWorld, Bessel Function Zeros

Cf. A115369 J'_0(1), A259616 J'_1(1), A259617 J'_2(1), A259618 J'_3(1), A259619 J'_4(1).

FindRoot[D[BesselJ[5, x], x] == 0, {x, 6}, WorkingPrecision -> 105] // Last // Last // RealDigits // First

A245461 Decimal expansion of constant in the Rayleigh criterion: first zero of J_1, divided by Pi.

Original entry on oeis.org

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

Offset: 1

Charles R Greathouse IV, Jul 22 2014

The angular resolution (diffraction limit), in radians, is this constant times the wavelength of the light used divided by the aperture diameter. That is: objects closer than this angle cannot be resolved as separate objects.
Note that the small angle approximation sin(theta) =~ theta can be used. Also, a commonly used approximation to this constant is 1.220. - John W. Nicholson, Jul 29 2014
For example, a radio telescope with a 300-meter aperture observing at the hydrogen line (21.106... cm) can distinguish objects at most 0.04916... degrees apart. - Charles R Greathouse IV, Mar 08 2018

			1.21966989126650445492653884746525517787935933077511...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
COSMOS - The SAO Encyclopedia of Astronomy, Rayleigh Criterion
Rene Kay, Todd Kay, Planet Seeker Interferometer (PSI), hal-02870885v2 [astro-ph.IM], 2020.
Wikipedia, Angular resolution
Wikipedia, Diffraction-limited system
Index entries for transcendental numbers

Cf. A115369, A221210.

RealDigits[ BesselJZero[1, 1]/Pi, 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2014 *)

PARI
```
solve(x=3,4,besselj(1,x))/Pi
```

besseljzero(1)/Pi \\ Charles R Greathouse IV, Aug 09 2022

Equals A115369/A000796.

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

Original entry on oeis.org

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

R. J. Mathar, Feb 21 2013

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 the first zero at 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.

Wikipedia, Airy disk

Cf. A245461.

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

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

cp's OEIS Frontend

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

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A115373 Decimal expansion of first zero of BesselJ(5,z).

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A259616 Decimal expansion of J'_1(1), the first root of the derivative of the Bessel function J_1.

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A259617 Decimal expansion of J'_2(1), the first root of the derivative of the Bessel function J_2.

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A259618 Decimal expansion of J'_3(1), the first root of the derivative of the Bessel function J_3.

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A259619 Decimal expansion of J'_4(1), the first root of the derivative of the Bessel function J_4.

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A259620 Decimal expansion of J'_5(1), the first root of the derivative of the Bessel function J_5.

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A245461 Decimal expansion of constant in the Rayleigh criterion: first zero of J_1, divided by Pi.

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