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Date: 19-1-2019
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Date: 8-3-2017
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When the index is real, the functions
,
,
, and
each have an infinite number of real zeros, all of which are simple with the possible exception of
. For nonnegative
, the
th positive zeros of these functions are denoted
,
,
, and
, respectively, except that
is typically counted as the first zero of
(Abramowitz and Stegun 1972, p. 370).
The first few roots of the Bessel function
are given in the following table for small nonnegative integer values of
and
. They can be found in the Wolfram Language using the command BesselJZero[n, k].
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1 | 2.4048 | 3.8317 | 5.1356 | 6.3802 | 7.5883 | 8.7715 |
2 | 5.5201 | 7.0156 | 8.4172 | 9.7610 | 11.0647 | 12.3386 |
3 | 8.6537 | 10.1735 | 11.6198 | 13.0152 | 14.3725 | 15.7002 |
4 | 11.7915 | 13.3237 | 14.7960 | 16.2235 | 17.6160 | 18.9801 |
5 | 14.9309 | 16.4706 | 17.9598 | 19.4094 | 20.8269 | 22.2178 |
The first few roots of the derivative of the Bessel function
are given in the following table for small nonnegative integer values of
and
. Versions of the Wolfram Language prior to 6 implemented these zeros as BesselJPrimeZeros[n, k] in the BesselZeros package which is now available for separate download (Wolfram Research). Note that contrary to Abramowitz and Stegun (1972, p. 370), the Wolfram Language defines the first zero of
to be approximately 3.8317 rather than zero.
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1 | 3.8317 | 1.8412 | 3.0542 | 4.2012 | 5.3175 | 6.4156 |
2 | 7.0156 | 5.3314 | 6.7061 | 8.0152 | 9.2824 | 10.5199 |
3 | 10.1735 | 8.5363 | 9.9695 | 11.3459 | 12.6819 | 13.9872 |
4 | 13.3237 | 11.7060 | 13.1704 | 14.5858 | 15.9641 | 17.3128 |
5 | 16.4706 | 14.8636 | 16.3475 | 17.7887 | 19.1960 | 20.5755 |
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Zeros." §9.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 370-374, 1972.
Goodwin, E. T. and Staton, J. "Table of ." Quart. J. Mech. Appl. Math. 1, 220-224, 1948.
Olver, F. W. J. (Ed.). "Zeros and Associated Values." Royal Society Mathematical Tables, Vol. 7: Bessel Functions. Cambridge, England: Cambridge University Press, 1960.
Wolfram Research. "Wolfram Language & System Documentation Center: Upgrading from NumericalMath BesselZeros." http://reference.wolfram.com/language/Compatibility/tutorial/NumericalMath/BesselZeros.html.
Wolfram Research. "Wolfram Library Archive: NumericalMath BesselZeros Legacy Standard Add-On Package." library.wolfram.com/infocenter/MathSource/6777.
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