Bessel Function Zeros
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].
 |  |  |  |  |  |  |
| 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.
 |  |  |  |  |  |  |
| 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 |
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.
Weisstein, Eric W. "Bessel Function Zeros." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BesselFunctionZeros.html
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Bessel functions
