The zeros of any cylinder function or its derivative are simple, with the possible exceptions of in the case of the functions, and in the case of the derivatives.
If is real, then , , , and , each have an infinite number of positive real zeros. All of these zeros are simple, provided that in the case of , and in the case of . When all of their zeros are simple, the th positive zeros of these functions are denoted by , , , and respectively, except that is counted as the first zero of . Since we have
When , the zeros interlace according to the inequalities
For an extension see Pálmai and Apagyi (2011).
The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function and the contiguous function . See also Elbert and Laforgia (1994).
When the zeros of are all real. If and is not an integer, then the number of complex zeros of is . If is odd, then two of these zeros lie on the imaginary axis.
If , then the zeros of are all real.
For information on the real double zeros of and when and , respectively, see Döring (1971) and Kerimov and Skorokhodov (1986). The latter reference also has information on double zeros of the second and third derivatives of and .
No two of the functions , , , have any common zeros other than ; see Watson (1944, §15.28).
If is a zero of the cylinder function
where is a parameter, then
If is a zero of , then
The parameter may be regarded as a continuous variable and , as functions , of . If and these functions are fixed by
The functions and are related to the inverses of the phase functions and defined in §10.18(i): if , then
For sign properties of the forward differences that are defined by
Some information on the distribution of and for real values of and is given in Muldoon and Spigler (1984).
Any positive zero of the cylinder function and any positive zero of such that are definable as continuous and increasing functions of :
where is defined in §10.25(ii).
In particular, , , , and are increasing functions of when . It is also true that the positive zeros and of and , respectively, are increasing functions of when , provided that in the latter case when .
and are decreasing functions of when for .
For bounds for the smallest real or purely imaginary zeros of when is real see Ismail and Muldoon (1995).
If is fixed, , and , then
where for , for . With , the right-hand side is the asymptotic expansion of for large .
where for , for , and for .
For the th positive zero of Wong and Lang (1990) gives the corresponding expansion
where if , and if . An error bound is included for the case .
As with fixed,
where is given by
As with fixed,
where is given by
In particular, with the notation as below,
Here , , , are the th negative zeros of , , , , respectively (§9.9), , , , are given by (10.21.25), (10.21.26), (10.21.30), and (10.21.31), with in the case of and , in the case of and , in the case of and , in the case of and .
For the first zeros rounded numerical values of the coefficients are given by
For numerical coefficients for see Olver (1951, Tables 3–6).
As the following four approximations hold uniformly for :
Here and denote respectively the zeros of the Airy function and its derivative ; see §9.9. Next, is the inverse of the function defined by (10.20.3). and are defined by (10.20.11) and (10.20.12) with . Lastly,
(Note: If the term in (10.21.43) is omitted, then the uniform character of the error term is destroyed.)
This subsection describes the distribution in of the zeros of the principal branches of the Bessel functions of the second and third kinds, and their derivatives, in the case when the order is a positive integer . For further information, including uniform asymptotic expansions, extensions to other branches of the functions and their derivatives, and extensions to half-integer values of the order, see Olver (1954). (There is an inaccuracy in Figures 11 and 14 in this reference. Each curve that represents an infinite string of nonreal zeros should be located on the opposite side of its straight line asymptote. This inaccuracy was repeated in Abramowitz and Stegun (1964, Figures 9.5 and 9.6). See Kerimov and Skorokhodov (1985a, b) and Figures 10.21.3–10.21.6.)
See also Cruz and Sesma (1982), Cruz et al. (1991), Kerimov and Skorokhodov (1984c, 1987, 1988), Kokologiannaki et al. (1992), and references supplied in §10.75(iii). For describing the distribution of complex zeros by methods based on the Liouville-Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013).
In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points are the boundaries of , that is, the eye-shaped domain depicted in Figure 10.20.3. These curves therefore intersect the imaginary axis at the points , where .
The first set of zeros of the principal value of are the points , , on the positive real axis (§10.21(i)). Secondly, there is a conjugate pair of infinite strings of zeros with asymptotes , where
Lastly, there are two conjugate sets, with zeros in each set, that are asymptotically close to the boundary of as . Figures 10.21.1, 10.21.3, and 10.21.5 plot the actual zeros for , and , respectively.
The zeros of have a similar pattern to those of .
The first set of zeros of the principal value of is an infinite string with asymptote , where
The zeros of have a similar pattern to those of . The zeros of and are the complex conjugates of the zeros of and , respectively.
Throughout this subsection we assume , , , and we denote by .
The zeros of the functions
are simple and the asymptotic expansion of the th positive zero as is given by
where, in the case of (10.21.48),
and, in the case of (10.21.49),
The asymptotic expansion of the large positive zeros (not necessarily the th) of the function
is given by (10.21.50), where
Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions.
The Riccati–Bessel functions are and . Except possibly for their zeros are the same as those of and , respectively. For information on the zeros of the derivatives of Riccati–Bessel functions, and also on zeros of their cross-products, see Boyer (1969). This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x).
For properties of the positive zeros of the function , with and real, see Landau (1999).