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.