Bessel Function Zeros
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Zeros." §9.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
4-3-2019
1692
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 |
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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