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10 Bessel FunctionsBessel and Hankel Functions

§10.21 Zeros


§10.21(i) Distribution

The zeros of any cylinder function or its derivative are simple, with the possible exceptions of z=0 in the case of the functions, and z=0,±ν in the case of the derivatives.

If ν is real, then Jν(z), Jν(z), Yν(z), and Yν(z), each have an infinite number of positive real zeros. All of these zeros are simple, provided that ν-1 in the case of Jν(z), and ν-12 in the case of Yν(z). When all of their zeros are simple, the mth positive zeros of these functions are denoted by jν,m, jν,m, yν,m, and yν,m respectively, except that z=0 is counted as the first zero of J0(z). Since J0(z)=-J1(z) we have

10.21.1 j0,1 =0,
j0,m =j1,m-1,

When ν0, the zeros interlace according to the inequalities

10.21.2 jν,1 <jν+1,1<jν,2<jν+1,2<jν,3<,
yν,1 <yν+1,1<yν,2<yν+1,2<yν,3<,
10.21.3 νjν,1<yν,1<yν,1<jν,1<jν,2<yν,2<.

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 𝒞ν(z) and the contiguous function 𝒞ν+1(z). See also Elbert and Laforgia (1994).

When ν-1 the zeros of Jν(z) are all real. If ν<-1 and ν is not an integer, then the number of complex zeros of Jν(z) is 2-ν. If -ν is odd, then two of these zeros lie on the imaginary axis.

If ν0, then the zeros of Jν(z) are all real.

For information on the real double zeros of Jν(z) and Yν(z) when ν<-1 and ν<-12, 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 Jν(z) and Yν(z).

No two of the functions J0(z), J1(z), J2(z),, have any common zeros other than z=0; see Watson (1944, §15.28).

§10.21(ii) Analytic Properties

If ρν is a zero of the cylinder function

10.21.4 𝒞ν(z)=Jν(z)cos(πt)+Yν(z)sin(πt),

where t is a parameter, then

10.21.5 𝒞ν(ρν)=𝒞ν-1(ρν)=-𝒞ν+1(ρν).

If σν is a zero of 𝒞ν(z), then

10.21.6 𝒞ν(σν)=σνν𝒞ν-1(σν)=σνν𝒞ν+1(σν).

The parameter t may be regarded as a continuous variable and ρν, σν as functions ρν(t), σν(t) of t. If ν0 and these functions are fixed by

10.21.7 ρν(0) =0,
σν(0) =jν,1,


10.21.8 jν,m =ρν(m),
yν,m =ρν(m-12),
10.21.9 jν,m =σν(m-1),
yν,m =σν(m-12),
10.21.10 𝒞ν(ρν) =(ρν2dρνdt)-12,
𝒞ν(σν) =(σν2-ν22σνdσνdt)-12,
10.21.11 2ρν2dρνdtd3ρνdt3-3ρν2(d2ρνdt2)2-4π2ρν2(dρνdt)2+(4ρν2+1-4ν2)(dρνdt)4=0.

The functions ρν(t) and σν(t) are related to the inverses of the phase functions θν(x) and ϕν(x) defined in §10.18(i): if ν0, then

10.21.12 θν(jν,m) =(m-12)π,
θν(yν,m) =(m-1)π,
10.21.13 ϕν(jν,m) =(m-12)π,
ϕν(yν,m) =mπ,

For sign properties of the forward differences that are defined by

10.21.14 Δρν(t) =ρν(t+1)-ρν(t),
Δ2ρν(t) =Δρν(t+1)-Δρν(t),,

when t=1,2,3,, and similarly for σν(t), see Lorch and Szegő (1963, 1964), Lorch et al. (1970, 1972), and Muldoon (1977).

Some information on the distribution of ρν(t) and σν(t) for real values of ν and t is given in Muldoon and Spigler (1984).

§10.21(iii) Infinite Products

10.21.15 Jν(z) =(12z)νΓ(ν+1)k=1(1-z2jν,k2),
10.21.16 Jν(z) =(12z)ν-12Γ(ν)k=1(1-z2jν,k2),

§10.21(iv) Monotonicity Properties

Any positive zero c of the cylinder function 𝒞ν(x) and any positive zero c of 𝒞ν(x) such that c>|ν| are definable as continuous and increasing functions of ν:

10.21.17 dcdν=2c0K0(2csinht)e-2νtdt,
10.21.18 dcdν=2cc2-ν20(c2cosh(2t)-ν2)K0(2csinht)e-2νtdt,

where K0 is defined in §10.25(ii).

In particular, jν,m, yν,m, jν,m, and yν,m are increasing functions of ν when ν0. It is also true that the positive zeros jν′′ and jν′′′ of Jν′′(x) and Jν′′′(x), respectively, are increasing functions of ν when ν>0, provided that in the latter case jν′′′>3 when 0<ν<1.

jν,m/ν and jν,m/ν are decreasing functions of ν when ν>0 for m=1,2,3,.

For further monotonicity properties see Elbert (2001), Lorch (1990, 1993, 1995), Lorch and Muldoon (2008), Lorch and Szegő (1990, 1995), and Muldoon (1981). For inequalities for zeros arising from monotonicity properties see Laforgia and Muldoon (1983).

§10.21(v) Inequalities

For bounds for the smallest real or purely imaginary zeros of Jν(x) when ν is real see Ismail and Muldoon (1995).

§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

If ν (0) is fixed, μ=4ν2, and m, then

10.21.19 jν,m,yν,ma-μ-18a-4(μ-1)(7μ-31)3(8a)3-32(μ-1)(83μ2-982μ+3779)15(8a)5-64(μ-1)(6949μ3-1 53855μ2+15 85743μ-62 77237)105(8a)7-,

where a=(m+12ν-14)π for jν,m, a=(m+12ν-34)π for yν,m. With a=(t+12ν-14)π, the right-hand side is the asymptotic expansion of ρν(t) for large t.

10.21.20 jν,m,yν,mb-μ+38b-4(7μ2+82μ-9)3(8b)3-32(83μ3+2075μ2-3039μ+3537)15(8b)5-64(6949μ4+2 96492μ3-12 48002μ2+74 14380μ-58 53627)105(8b)7-,

where b=(m+12ν-34)π for jν,m, b=(m+12ν-14)π for yν,m, and b=(t+12ν+14)π for σν(t).

For the next three terms in (10.21.19) and the next two terms in (10.21.20) see Bickley et al. (1952, p. xxxvii) or Olver (1960, pp. xvii–xviii).

For error bounds see Wong and Lang (1990), Wong (1995), and Elbert and Laforgia (2000). See also Laforgia (1979).

For the mth positive zero jν,m′′ of Jν′′(x) Wong and Lang (1990) gives the corresponding expansion

10.21.21 jν,m′′c-μ+78c-28μ2+424μ+17243(8c)3-,

where c=(m+12ν-14)π if 0<ν<1, and c=(m+12ν-54)π if ν>1. An error bound is included for the case ν32.

§10.21(vii) Asymptotic Expansions for Large Order

Let 𝒞ν(x), ρν(t), and σν(t) be defined as in §10.21(ii) and M(x), θ(x), N(x), and ϕ(x) denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8.

As ν with t (>0) fixed,

10.21.22 ρν(t)νk=0αkν2k/3,
10.21.23 𝒞ν(ρν(t))(2/ν)23πM(-213α)k=0βkν2k/3,

where α is given by

10.21.24 θ(-213α)=πt,


10.21.25 α0 =1,
α1 =α,
α2 =310α2,
α3 =-1350α3+170,
α4 =-47963000α4-13150α,
α5 =2023180 85000α5-5511 61700α2,
10.21.26 β0 =1,
β1 =-45α,
β2 =1835α2,
β3 =-88315α3-111575,
β4 =795866 06375α4+98246 06375α.

As ν with t (>-16) fixed,

10.21.27 σν(t)νk=0αkν2k/3,
10.21.28 𝒞ν(σν(t))(2/ν)13πN(-213α)k=0βkν2k/3,

where α is given by

10.21.29 ϕ(-213α)=πt,


10.21.30 α0 =1,
α1 =α,
α2 =310α2-110α-1,
α3 =-1350α3-125-1200α-3,
α4 =-47963000α4+50931500α+11500α-2-12000α-5,
10.21.31 β0 =1,
β1 =-15α,
β2 =9350α2+1100α-1,
β3 =8915750α3-474500+13000α-3.

In particular, with the notation as below,

10.21.32 jν,mνk=0αkν2k/3,
10.21.33 yν,mνk=0αkν2k/3,
10.21.34 Jν(jν,m)(-1)m(2/ν)23πM(am)k=0βkν2k/3,
10.21.35 Yν(yν,m)(-1)m-1(2/ν)23πM(bm)k=0βkν2k/3,


10.21.36 jν,mνk=0αkν2k/3,
10.21.37 yν,mνk=0αkν2k/3,
10.21.38 Jν(jν,m)(-1)m-1(2/ν)13πN(am)k=0βkν2k/3,
10.21.39 Yν(yν,m)(-1)m-1(2/ν)13πN(bm)k=0βkν2k/3.

Here am, bm, am, bm are the mth negative zeros of Ai(x), Bi(x), Ai(x), Bi(x), respectively (§9.9), αk, βk, αk, βk are given by (10.21.25), (10.21.26), (10.21.30), and (10.21.31), with α=-2-13am in the case of jν,m and Jν(jν,m), α=-2-13bm in the case of yν,m and Yν(yν,m), α=-2-13am in the case of jν,m and Jν(jν,m), α=-2-13bm in the case of yν,m and Yν(yν,m).

For error bounds for (10.21.32) see Qu and Wong (1999); for (10.21.36) and (10.21.37) see Elbert and Laforgia (1997). See also Spigler (1980).

For the first zeros rounded numerical values of the coefficients are given by

10.21.40 jν,1 ν+1.85575 71ν13+1.03315 0ν-13-0.00397ν-1-0.0908ν-53+0.043ν-73+,
yν,1 ν+0.93157 68ν13+0.26035 1ν-13+0.01198ν-1-0.0060ν-53-0.001ν-73+,
Jν(jν,1) -1.11310 28ν-23÷(1+1.48460 6ν-23+0.43294ν-43-0.1943ν-2+0.019ν-83+),
Yν(yν,1) 0.95554 86ν-23÷(1+0.74526 1ν-23+0.10910ν-43-0.0185ν-2-0.003ν-83+),
jν,1 ν+0.80861 65ν13+0.07249 0ν-13-0.05097ν-1+0.0094ν-53+,
yν,1 ν+1.82109 80ν13+0.94000 7ν-13-0.05808ν-1-0.0540ν-53+.
Jν(jν,1) 0.67488 51ν-13(1-0.16172 3ν-23+0.02918ν-43-0.0068ν-2+),
Yν(yν,1) 0.57319 40ν-13(1-0.36422 0ν-23+0.09077ν-43+0.0237ν-2+).

For numerical coefficients for m=2,3,4,5 see Olver (1951, Tables 3–6).

The expansions (10.21.32)–(10.21.39) become progressively weaker as m increases. The approximations that follow in §10.21(viii) do not suffer from this drawback.

§10.21(viii) Uniform Asymptotic Approximations for Large Order

As ν the following four approximations hold uniformly for m=1,2,:

10.21.41 jν,m=νz(ζ)+z(ζ)(h(ζ))2B0(ζ)2ν+O(1ν3),
10.21.42 Jν(jν,m)=-2ν23Ai(am)z(ζ)h(ζ)(1+O(1ν2)),
10.21.43 jν,m=νz(ζ)+z(ζ)(h(ζ))2C0(ζ)2ζν+O(1ν),
10.21.44 Jν(jν,m)=h(ζ)Ai(am)ν13(1+O(1ν43)),

Here am and am denote respectively the zeros of the Airy function Ai(z) and its derivative Ai(z); see §9.9. Next, z(ζ) is the inverse of the function ζ=ζ(z) defined by (10.20.3). B0(ζ) and C0(ζ) are defined by (10.20.11) and (10.20.12) with k=0. Lastly,

10.21.45 h(ζ)=(4ζ/(1-z2))14.

(Note: If the term z(ζ)(h(ζ))2C0(ζ)/(2ζν) in (10.21.43) is omitted, then the uniform character of the error term O(1/ν) is destroyed.)

Corresponding uniform approximations for yν,m, Yν(yν,m), yν,m, and Yν(yν,m), are obtained from (10.21.41)–(10.21.44) by changing the symbols j, J, Ai, Ai, am, and am to y, Y, -Bi, -Bi, bm, and bm, respectively.

For derivations and further information, including extensions to uniform asymptotic expansions, see Olver (1954, 1960). The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions.

§10.21(ix) Complex Zeros

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 n. 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 10.21.310.21.6.)

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).

Zeros of Yn(nz) and Yn(nz)

In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points ±1 are the boundaries of K, that is, the eye-shaped domain depicted in Figure 10.20.3. These curves therefore intersect the imaginary axis at the points z=±ic, where c=0.66274.

The first set of zeros of the principal value of Yn(nz) are the points z=yn,m/n, m=1,2,, on the positive real axis (§10.21(i)). Secondly, there is a conjugate pair of infinite strings of zeros with asymptotes z=±ia/n, where

10.21.46 a=12ln3=0.54931.

Lastly, there are two conjugate sets, with n zeros in each set, that are asymptotically close to the boundary of K as n. Figures 10.21.1, 10.21.3, and 10.21.5 plot the actual zeros for n=1,5, and 10, respectively.

The zeros of Yn(nz) have a similar pattern to those of Yn(nz).

See accompanying text
Figure 10.21.1: Zeros of Yn(nz) in |phz|π. Case n=1, -1.6z2.6. Magnify
See accompanying text
Figure 10.21.2: Zeros of Hn(1)(nz) in |phz|π. Case n=1, -2.8z1.4. Magnify
See accompanying text
Figure 10.21.3: Zeros of Yn(nz) in |phz|π. Case n=5, -2.6z1.6. Magnify
See accompanying text
Figure 10.21.4: Zeros of Hn(1)(nz) in |phz|π. Case n=5, -2.6z1.6. Magnify
See accompanying text
Figure 10.21.5: Zeros of Yn(nz) in |phz|π. Case n=10, -2.3z1.9. Magnify
See accompanying text
Figure 10.21.6: Zeros of Hn(1)(nz) in |phz|π. Case n=10, -2.3z1.9. Magnify

Zeros of Hn(1)(nz), Hn(2)(nz), Hn(1)(nz), Hn(2)(nz)

In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points ±1 is the lower boundary of K.

The first set of zeros of the principal value of Hn(1)(nz) is an infinite string with asymptote z=-id/n, where

10.21.47 d=12ln2=0.34657.

The only other set comprises n zeros that are asymptotically close to the lower boundary of K as n. Figures 10.21.2, 10.21.4, and 10.21.6 plot the actual zeros for n=1,5, and 10, respectively.

The zeros of Hn(1)(nz) have a similar pattern to those of Hn(1)(nz). The zeros of Hn(2)(nz) and Hn(2)(nz) are the complex conjugates of the zeros of Hn(1)(nz) and Hn(1)(nz), respectively.

Zeros of J0(z)-iJ1(z) and Jn(z)-iJn+1(z)

For information see Synolakis (1988), MacDonald (1989, 1997), and Ikebe et al. (1993).

§10.21(x) Cross-Products

Throughout this subsection we assume ν0, x>0, λ>1, and we denote 4ν2 by μ.

The zeros of the functions

10.21.48 Jν(x)Yν(λx)-Yν(x)Jν(λx)


10.21.49 Jν(x)Yν(λx)-Yν(x)Jν(λx)

are simple and the asymptotic expansion of the mth positive zero as m is given by

10.21.50 α+pα+q-p2α3+r-4pq+2p3α5+,

where, in the case of (10.21.48),

10.21.51 α =mπλ-1,
p =μ-18λ,
q =(μ-1)(μ-25)(λ3-1)6(4λ)3(λ-1),
r =(μ-1)(μ2-114μ+1073)(λ5-1)5(4λ)5(λ-1),

and, in the case of (10.21.49),

10.21.52 α =(m-1)πλ-1,
p =μ+38λ,
q =(μ2+46μ-63)(λ3-1)6(4λ)3(λ-1),
r =(μ3+185μ2-2053μ+1899)(λ5-1)5(4λ)5(λ-1).

The asymptotic expansion of the large positive zeros (not necessarily the mth) of the function

10.21.53 Jν(x)Yν(λx)-Yν(x)Jν(λx)

is given by (10.21.50), where

10.21.54 α =(m-12)πλ-1,
p =(μ+3)λ-(μ-1)8λ(λ-1),
q =(μ2+46μ-63)λ3-(μ-1)(μ-25)6(4λ)3(λ-1),
r =(μ3+185μ2-2053μ+1899)λ5-(μ-1)(μ2-114μ+1073)5(4λ)5(λ-1).

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.

For further information see Cochran (1963, 1964, 1966a, 1966b), Kalähne (1907), Martinek et al. (1966), Muldoon (1979), and Salchev and Popov (1976).

§10.21(xi) Riccati–Bessel Functions

The Riccati–Bessel functions are (12πx)12Jν(x) and (12πx)12Yν(x). Except possibly for x=0 their zeros are the same as those of Jν(x) and Yν(x), 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).

§10.21(xii) Zeros of αJν(x)+xJν(x)

For properties of the positive zeros of the function αJν(x)+xJν(x), with α and ν real, see Landau (1999).

§10.21(xiii) Rayleigh Function

The Rayleigh function σn(ν) is defined by

10.21.55 σn(ν)=m=1(jν,m)-2n,

For properties, computation, and generalizations see Kapitsa (1951b), Kerimov (1999, 2008), and Gupta and Muldoon (2000). See also Watson (1944, §§15.5, 15.51).

§10.21(xiv) ν-Zeros

For information on zeros of Bessel and Hankel functions as functions of the order, see Cochran (1965), Cochran and Hoffspiegel (1970), Hethcote (1970), Conde and Kalla (1979), and Sandström and Ackrén (2007).