§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 $z=0,\pm\nu$ in the case of the derivatives.

If $\nu$ is real, then $J_{\nu}\left(z\right)$, $J_{\nu}'\left(z\right)$, $Y_{\nu}\left(z\right)$, and $Y_{\nu}'\left(z\right)$, each have an infinite number of positive real zeros. All of these zeros are simple, provided that $\nu\geq-1$ in the case of $J_{\nu}'\left(z\right)$, and $\nu\geq-\tfrac{1}{2}$ in the case of $Y_{\nu}'\left(z\right)$. When all of their zeros are simple, the $m$th positive zeros of these functions are denoted by $j_{\nu,m}$, ${j^{\prime}_{\nu,m}}$, $y_{\nu,m}$, and ${y^{\prime}_{\nu,m}}$ respectively, except that $z=0$ is counted as the first zero of $J_{0}'\left(z\right)$. Since $J_{0}'\left(z\right)=-J_{1}\left(z\right)$ we have

 10.21.1 $\displaystyle{j^{\prime}_{0,1}}$ $\displaystyle=0,$ $\displaystyle{j^{\prime}_{0,m}}$ $\displaystyle=j_{1,m-1},$ $m=2,3,\dotsc$.

When $\nu\geq 0$, the zeros interlace according to the inequalities

 10.21.2 $\displaystyle j_{\nu,1}$ $\displaystyle $\displaystyle y_{\nu,1}$ $\displaystyle ⓘ Symbols: $j_{\NVar{\nu},\NVar{m}}$: zeros of the Bessel function $J_{\nu}\left(x\right)$, $y_{\NVar{\nu},\NVar{m}}$: zeros of the Bessel function $Y_{\nu}\left(x\right)$ and $\nu$: complex parameter A&S Ref: 9.5.2 Referenced by: §10.21(ii) Permalink: http://dlmf.nist.gov/10.21.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.21(i), §10.21 and Ch.10

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 $\mathscr{C}_{\nu}\left(z\right)$ and the contiguous function $\mathscr{C}_{\nu+1}\left(z\right)$. See also Elbert and Laforgia (1994).

When $\nu\geq-1$ the zeros of $J_{\nu}\left(z\right)$ are all real. If $\nu<-1$ and $\nu$ is not an integer, then the number of complex zeros of $J_{\nu}\left(z\right)$ is $2\left\lfloor-\nu\right\rfloor$. If $\left\lfloor-\nu\right\rfloor$ is odd, then two of these zeros lie on the imaginary axis.

If $\nu\geq 0$, then the zeros of $J_{\nu}'\left(z\right)$ are all real.

For information on the real double zeros of $J_{\nu}'\left(z\right)$ and $Y_{\nu}'\left(z\right)$ when $\nu<-1$ and $\nu<-\tfrac{1}{2}$, 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 $J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$.

No two of the functions $J_{0}\left(z\right)$, $J_{1}\left(z\right)$, $J_{2}\left(z\right),\dotsc$, have any common zeros other than $z=0$; see Watson (1944, §15.28).

§10.21(ii) Analytic Properties

If $\rho_{\nu}$ is a zero of the cylinder function

 10.21.4 $\mathscr{C}_{\nu}\left(z\right)=J_{\nu}\left(z\right)\cos\left(\pi t\right)+Y_% {\nu}\left(z\right)\sin\left(\pi t\right),$

where $t$ is a parameter, then

 10.21.5 $\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)=\mathscr{C}_{\nu-1}\left(\rho_{\nu}% \right)=-\mathscr{C}_{\nu+1}\left(\rho_{\nu}\right).$ ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $\nu$: complex parameter and $\rho_{\nu}(t)$: zero of cylinder function A&S Ref: 9.5.4 Referenced by: §10.21(ii) Permalink: http://dlmf.nist.gov/10.21.E5 Encodings: TeX, pMML, png See also: Annotations for §10.21(ii), §10.21 and Ch.10

If $\sigma_{\nu}$ is a zero of $\mathscr{C}_{\nu}'\left(z\right)$, then

 10.21.6 $\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}% _{\nu-1}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}_{\nu+1}% \left(\sigma_{\nu}\right).$ ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $\nu$: complex parameter and $\sigma_{\nu}(t)$: zero of cylinder function A&S Ref: 9.5.5 Referenced by: §10.21(ii) Permalink: http://dlmf.nist.gov/10.21.E6 Encodings: TeX, pMML, png See also: Annotations for §10.21(ii), §10.21 and Ch.10

The parameter $t$ may be regarded as a continuous variable and $\rho_{\nu}$, $\sigma_{\nu}$ as functions $\rho_{\nu}(t)$, $\sigma_{\nu}(t)$ of $t$. If $\nu\geq 0$ and these functions are fixed by

 10.21.7 $\displaystyle\rho_{\nu}(0)$ $\displaystyle=0,$ $\displaystyle\sigma_{\nu}(0)$ $\displaystyle={j^{\prime}_{\nu,1}},$

then

 10.21.8 $\displaystyle j_{\nu,m}$ $\displaystyle=\rho_{\nu}(m),$ $\displaystyle y_{\nu,m}$ $\displaystyle=\rho_{\nu}(m-\tfrac{1}{2})$, $m=1,2,\dotsc$,
 10.21.9 $\displaystyle{j^{\prime}_{\nu,m}}$ $\displaystyle=\sigma_{\nu}(m-1),$ $\displaystyle{y^{\prime}_{\nu,m}}$ $\displaystyle=\sigma_{\nu}(m-\tfrac{1}{2})$, $m=1,2,\dotsc$.
 10.21.10 $\displaystyle\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)$ $\displaystyle=\left(\frac{\rho_{\nu}}{2}\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}% t}\right)^{-\frac{1}{2}},$ $\displaystyle\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)$ $\displaystyle=\left(\frac{\sigma_{\nu}^{2}-\nu^{2}}{2\sigma_{\nu}}\frac{% \mathrm{d}\sigma_{\nu}}{\mathrm{d}t}\right)^{-\frac{1}{2}},$
 10.21.11 $2\rho_{\nu}^{2}\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\frac{{\mathrm{d}}^{3}% \rho_{\nu}}{{\mathrm{d}t}^{3}}-3\rho_{\nu}^{2}\*\left(\frac{{\mathrm{d}}^{2}% \rho_{\nu}}{{\mathrm{d}t}^{2}}\right)^{2}-4\pi^{2}\rho_{\nu}^{2}\*\left(\frac{% \mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{2}+(4\rho_{\nu}^{2}+1-4\nu^{2})% \left(\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{4}=0.$

The functions $\rho_{\nu}(t)$ and $\sigma_{\nu}(t)$ are related to the inverses of the phase functions $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ defined in §10.18(i): if $\nu\geq 0$, then

 10.21.12 $\displaystyle\theta_{\nu}\left(j_{\nu,m}\right)$ $\displaystyle=(m-\tfrac{1}{2})\pi,$ $\displaystyle\theta_{\nu}\left(y_{\nu,m}\right)$ $\displaystyle=(m-1)\pi,$ $m=1,2,\dotsc$,
 10.21.13 $\displaystyle\phi_{\nu}\left({j^{\prime}_{\nu,m}}\right)$ $\displaystyle=(m-\tfrac{1}{2})\pi,$ $\displaystyle\phi_{\nu}\left({y^{\prime}_{\nu,m}}\right)$ $\displaystyle=m\pi,$ $m=1,2,\dotsc$.

For sign properties of the forward differences that are defined by

 10.21.14 $\displaystyle\Delta\rho_{\nu}(t)$ $\displaystyle=\rho_{\nu}(t+1)-\rho_{\nu}(t),$ $\displaystyle\Delta^{2}\rho_{\nu}(t)$ $\displaystyle=\Delta\rho_{\nu}(t+1)-\Delta\rho_{\nu}(t),\dotsc,$ ⓘ Symbols: $\Delta$: forward difference operator, $\nu$: complex parameter and $\rho_{\nu}(t)$: zero of cylinder function Permalink: http://dlmf.nist.gov/10.21.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.21(ii), §10.21 and Ch.10

when $t=1,2,3,\dotsc$, and similarly for $\sigma_{\nu}(t)$, see Lorch and Szegő (1963, 1964), Lorch et al. (1970, 1972), and Muldoon (1977).

Some information on the distribution of $\rho_{\nu}(t)$ and $\sigma_{\nu}(t)$ for real values of $\nu$ and $t$ is given in Muldoon and Spigler (1984).

§10.21(iii) Infinite Products

 10.21.15 $\displaystyle J_{\nu}\left(z\right)$ $\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}\prod_{k=1% }^{\infty}\left(1-\frac{z^{2}}{{j_{\nu,k}^{2}}}\right),$ $\nu\geq 0$, 10.21.16 $\displaystyle J_{\nu}'\left(z\right)$ $\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu-1}}{2\Gamma\left(\nu\right)}\prod_{k=% 1}^{\infty}\left(1-\frac{z^{2}}{{{j^{\prime}_{\nu,k}}^{\mspace{-16.0mu }2}}}% \right),$ $\nu>0$.

§10.21(iv) Monotonicity Properties

Any positive zero $c$ of the cylinder function $\mathscr{C}_{\nu}\left(x\right)$ and any positive zero $c^{\prime}$ of $\mathscr{C}_{\nu}'\left(x\right)$ such that $c^{\prime}>|\nu|$ are definable as continuous and increasing functions of $\nu$:

 10.21.17 $\frac{\mathrm{d}c}{\mathrm{d}\nu}=2c\int_{0}^{\infty}K_{0}\left(2c\sinh t% \right)e^{-2\nu t}\mathrm{d}t,$
 10.21.18 $\frac{\mathrm{d}c^{\prime}}{\mathrm{d}\nu}=\frac{2c^{\prime}}{{c^{\prime}}^{2}% -\nu^{2}}\*\int_{0}^{\infty}({c^{\prime}}^{2}\cosh\left(2t\right)-\nu^{2})\*K_% {0}\left(2c^{\prime}\sinh t\right)e^{-2\nu t}\mathrm{d}t,$

where $K_{0}$ is defined in §10.25(ii).

In particular, $j_{\nu,m}$, $y_{\nu,m}$, $j_{\nu,m}'$, and $y_{\nu,m}'$ are increasing functions of $\nu$ when $\nu\geq 0$. It is also true that the positive zeros $j^{\prime\prime}_{\nu}$ and $j^{\prime\prime\prime}_{\nu}$ of $J_{\nu}''\left(x\right)$ and $J_{\nu}'''\left(x\right)$, respectively, are increasing functions of $\nu$ when $\nu>0$, provided that in the latter case $j^{\prime\prime\prime}_{\nu}>\sqrt{3}$ when $0<\nu<1$.

$j_{\nu,m}/\nu$ and $j_{\nu,m}'/\nu$ are decreasing functions of $\nu$ when $\nu>0$ for $m=1,2,3,\dotsc$.

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 $J_{\nu}\left(x\right)$ when $\nu$ is real see Ismail and Muldoon (1995).

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

If $\nu$ $(\geq 0)$ is fixed, $\mu=4\nu^{2}$, and $m\to\infty$, then

 10.21.19 $j_{\nu,m},y_{\nu,m}\sim a-\frac{\mu-1}{8a}-\frac{4(\mu-1)(7\mu-31)}{3(8a)^{3}}% -\frac{32(\mu-1)(83\mu^{2}-982\mu+3779)}{15(8a)^{5}}-\frac{64(\mu-1)(6949\mu^{% 3}-1\;53855\mu^{2}+15\;85743\mu-62\;77237)}{105(8a)^{7}}-\dotsb,$

where $a=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi$ for $j_{\nu,m}$, $a=(m+\tfrac{1}{2}\nu-\tfrac{3}{4})\pi$ for $y_{\nu,m}$. With $a=(t+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi$, the right-hand side is the asymptotic expansion of $\rho_{\nu}(t)$ for large $t$.

 10.21.20 $j_{\nu,m}',y_{\nu,m}'\sim b-\frac{\mu+3}{8b}-\frac{4(7\mu^{2}+82\mu-9)}{3(8b)^% {3}}-\frac{32(83\mu^{3}+2075\mu^{2}-3039\mu+3537)}{15(8b)^{5}}-\frac{64(6949% \mu^{4}+2\;96492\mu^{3}-12\;48002\mu^{2}+74\;14380\mu-58\;53627)}{105(8b)^{7}}% -\dotsb,$

where $b=(m+\tfrac{1}{2}\nu-\tfrac{3}{4})\pi$ for $j_{\nu,m}'$, $b=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi$ for $y_{\nu,m}'$, and $b=(t+\tfrac{1}{2}\nu+\tfrac{1}{4})\pi$ for $\sigma_{\nu}(t)$.

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 $m$th positive zero $j^{\prime\prime}_{\nu,m}$ of $J_{\nu}''\left(x\right)$ Wong and Lang (1990) gives the corresponding expansion

 10.21.21 $j^{\prime\prime}_{\nu,m}\sim c-\frac{\mu+7}{8c}-\frac{28\mu^{2}+424\mu+1724}{3% (8c)^{3}}-\dotsb,$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $m$: integer, $\nu$: complex parameter and $c$: zero of cylinder function Permalink: http://dlmf.nist.gov/10.21.E21 Encodings: TeX, pMML, png See also: Annotations for §10.21(vi), §10.21 and Ch.10

where $c=(m+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi$ if $0<\nu<1$, and $c=(m+\tfrac{1}{2}\nu-\tfrac{5}{4})\pi$ if $\nu>1$. An error bound is included for the case $\nu\geq\tfrac{3}{2}$.

§10.21(vii) Asymptotic Expansions for Large Order

Let $\mathscr{C}_{\nu}\left(x\right)$, $\rho_{\nu}(t)$, and $\sigma_{\nu}(t)$ be defined as in §10.21(ii) and $M\left(x\right)$, $\theta\left(x\right)$, $N\left(x\right)$, and $\phi\left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8.

As $\nu\to\infty$ with $t$ $(>0)$ fixed,

 10.21.22 $\rho_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$
 10.21.23 $\mathscr{C}_{\nu}'\left(\rho_{\nu}(t)\right)\sim\frac{(2/\nu)^{\frac{2}{3}}}{% \pi M\left(-2^{\frac{1}{3}}\alpha\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{% \nu^{2k/3}},$

where $\alpha$ is given by

 10.21.24 $\theta\left(-2^{\frac{1}{3}}\alpha\right)=\pi t,$ ⓘ Defines: $\alpha$: coefficients (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\theta\left(\NVar{z}\right)$: Airy phase function Permalink: http://dlmf.nist.gov/10.21.E24 Encodings: TeX, pMML, png See also: Annotations for §10.21(vii), §10.21 and Ch.10

and

 10.21.25 $\displaystyle\alpha_{0}$ $\displaystyle=1,$ $\displaystyle\alpha_{1}$ $\displaystyle=\alpha,$ $\displaystyle\alpha_{2}$ $\displaystyle=\tfrac{3}{10}\alpha^{2},$ $\displaystyle\alpha_{3}$ $\displaystyle=-\tfrac{1}{350}\alpha^{3}+\tfrac{1}{70},$ $\displaystyle\alpha_{4}$ $\displaystyle=-\tfrac{479}{63000}\alpha^{4}-\tfrac{1}{3150}\alpha,$ $\displaystyle\alpha_{5}$ $\displaystyle=\tfrac{20231}{80\;85000}\alpha^{5}-\tfrac{551}{1\;61700}\alpha^{% 2},$ ⓘ Symbols: $\alpha$: coefficients Referenced by: §10.21(vii) Permalink: http://dlmf.nist.gov/10.21.E25 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §10.21(vii), §10.21 and Ch.10
 10.21.26 $\displaystyle\beta_{0}$ $\displaystyle=1,$ $\displaystyle\beta_{1}$ $\displaystyle=-\tfrac{4}{5}\alpha,$ $\displaystyle\beta_{2}$ $\displaystyle=\tfrac{18}{35}\alpha^{2},$ $\displaystyle\beta_{3}$ $\displaystyle=-\tfrac{88}{315}\alpha^{3}-\tfrac{11}{1575},$ $\displaystyle\beta_{4}$ $\displaystyle=\tfrac{79586}{6\;06375}\alpha^{4}+\tfrac{9824}{6\;06375}\alpha.$ ⓘ Defines: $\beta$: coefficients (locally) Symbols: $\alpha$: coefficients Referenced by: §10.21(vii) Permalink: http://dlmf.nist.gov/10.21.E26 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §10.21(vii), §10.21 and Ch.10

As $\nu\to\infty$ with $t$ $(>-\tfrac{1}{6})$ fixed,

 10.21.27 $\sigma_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$
 10.21.28 $\mathscr{C}_{\nu}\left(\sigma_{\nu}(t)\right)\sim\frac{(2/\nu)^{\frac{1}{3}}}{% \pi N\left(-2^{\frac{1}{3}}\alpha^{\prime}\right)}\sum_{k=0}^{\infty}\frac{% \beta^{\prime}_{k}}{\nu^{2k/3}},$

where $\alpha^{\prime}$ is given by

 10.21.29 $\phi\left(-2^{\frac{1}{3}}\alpha^{\prime}\right)=\pi t,$ ⓘ Defines: $\alpha^{\prime}$: coefficients (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\phi\left(\NVar{z}\right)$: Airy phase function Permalink: http://dlmf.nist.gov/10.21.E29 Encodings: TeX, pMML, png See also: Annotations for §10.21(vii), §10.21 and Ch.10

and

 10.21.30 $\displaystyle\alpha^{\prime}_{0}$ $\displaystyle=1,$ $\displaystyle\alpha^{\prime}_{1}$ $\displaystyle=\alpha^{\prime},$ $\displaystyle\alpha^{\prime}_{2}$ $\displaystyle=\tfrac{3}{10}{\alpha^{\prime}}^{2}-\tfrac{1}{10}{\alpha^{\prime}% }^{-1},$ $\displaystyle\alpha^{\prime}_{3}$ $\displaystyle=-\tfrac{1}{350}{\alpha^{\prime}}^{3}-\tfrac{1}{25}-\tfrac{1}{200% }{\alpha^{\prime}}^{-3},$ $\displaystyle\alpha^{\prime}_{4}$ $\displaystyle=-\tfrac{479}{63000}{\alpha^{\prime}}^{4}+\tfrac{509}{31500}% \alpha^{\prime}+\tfrac{1}{1500}{\alpha^{\prime}}^{-2}-\tfrac{1}{2000}{\alpha^{% \prime}}^{-5},$ ⓘ Symbols: $\alpha^{\prime}$: coefficients Referenced by: §10.21(vii) Permalink: http://dlmf.nist.gov/10.21.E30 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §10.21(vii), §10.21 and Ch.10
 10.21.31 $\displaystyle\beta^{\prime}_{0}$ $\displaystyle=1,$ $\displaystyle\beta^{\prime}_{1}$ $\displaystyle=-\tfrac{1}{5}\alpha^{\prime},$ $\displaystyle\beta^{\prime}_{2}$ $\displaystyle=\tfrac{9}{350}{\alpha^{\prime}}^{2}+\tfrac{1}{100}{\alpha^{% \prime}}^{-1},$ $\displaystyle\beta^{\prime}_{3}$ $\displaystyle=\tfrac{89}{15750}{\alpha^{\prime}}^{3}-\tfrac{47}{4500}+\tfrac{1% }{3000}{\alpha^{\prime}}^{-3}.$ ⓘ Symbols: $\beta$: coefficients and $\alpha^{\prime}$: coefficients Referenced by: §10.21(vii) Permalink: http://dlmf.nist.gov/10.21.E31 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.21(vii), §10.21 and Ch.10

In particular, with the notation as below,

 10.21.32 $j_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$
 10.21.33 $y_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$
 10.21.34 $J_{\nu}'\left(j_{\nu,m}\right)\sim(-1)^{m}\frac{(2/\nu)^{\frac{2}{3}}}{\pi M% \left(a_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{\nu^{2k/3}},$
 10.21.35 $Y_{\nu}'\left(y_{\nu,m}\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{2}{3}}}{\pi M% \left(b_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta_{k}}{\nu^{2k/3}},$

and

 10.21.36 $j_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$
 10.21.37 $y_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$
 10.21.38 $J_{\nu}\left(j_{\nu,m}'\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{1}{3}}}{\pi N% \left(a^{\prime}_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta^{\prime}_{k}}{\nu^{% 2k/3}},$
 10.21.39 $Y_{\nu}\left(y_{\nu,m}'\right)\sim(-1)^{m-1}\frac{(2/\nu)^{\frac{1}{3}}}{\pi N% \left(b^{\prime}_{m}\right)}\sum_{k=0}^{\infty}\frac{\beta^{\prime}_{k}}{\nu^{% 2k/3}}.$

Here $a_{m}$, $b_{m}$, $a^{\prime}_{m}$, $b^{\prime}_{m}$ are the $m$th negative zeros of $\mathrm{Ai}\left(x\right)$, $\mathrm{Bi}\left(x\right)$, $\mathrm{Ai}'\left(x\right)$, $\mathrm{Bi}'\left(x\right)$, respectively (§9.9), $\alpha_{k}$, $\beta_{k}$, $\alpha^{\prime}_{k}$, $\beta^{\prime}_{k}$ are given by (10.21.25), (10.21.26), (10.21.30), and (10.21.31), with $\alpha=-2^{-\frac{1}{3}}a_{m}$ in the case of $j_{\nu,m}$ and $J_{\nu}'\left(j_{\nu,m}\right)$, $\alpha=-2^{-\frac{1}{3}}b_{m}$ in the case of $y_{\nu,m}$ and $Y_{\nu}'\left(y_{\nu,m}\right)$, $\alpha^{\prime}=-2^{-\frac{1}{3}}a^{\prime}_{m}$ in the case of $j_{\nu,m}'$ and $J_{\nu}\left(j_{\nu,m}'\right)$, $\alpha^{\prime}=-2^{-\frac{1}{3}}b^{\prime}_{m}$ in the case of $y_{\nu,m}'$ and $Y_{\nu}\left(y_{\nu,m}'\right)$.

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 $\displaystyle j_{\nu,1}$ $\displaystyle\sim\nu+1.85575\;71\nu^{\frac{1}{3}}+1.03315\;0\nu^{-\frac{1}{3}}% -0.00397\nu^{-1}-0.0908\nu^{-\frac{5}{3}}+0.043\nu^{-\frac{7}{3}}+\dotsb,$ $\displaystyle y_{\nu,1}$ $\displaystyle\sim\nu+0.93157\;68\nu^{\frac{1}{3}}+0.26035\;1\nu^{-\frac{1}{3}}% +0.01198\nu^{-1}-0.0060\nu^{-\frac{5}{3}}-0.001\nu^{-\frac{7}{3}}+\dotsb,$ $\displaystyle J_{\nu}'\left(j_{\nu,1}\right)$ $\displaystyle\sim-1.11310\;28\nu^{-\frac{2}{3}}\div(1+1.48460\;6\nu^{-\frac{2}% {3}}+0.43294\nu^{-\frac{4}{3}}-0.1943\nu^{-2}+0.019\nu^{-\frac{8}{3}}+\cdots),$ $\displaystyle Y_{\nu}'\left(y_{\nu,1}\right)$ $\displaystyle\sim 0.95554\;86\nu^{-\frac{2}{3}}\div(1+0.74526\;1\nu^{-\frac{2}% {3}}+0.10910\nu^{-\frac{4}{3}}-0.0185\nu^{-2}-0.003\nu^{-\frac{8}{3}}+\cdots),$ $\displaystyle{j^{\prime}_{\nu,1}}$ $\displaystyle\sim\nu+0.80861\;65\nu^{\frac{1}{3}}+0.07249\;0\nu^{-\frac{1}{3}}% -0.05097\nu^{-1}+0.0094\nu^{-\frac{5}{3}}+\dotsb,$ $\displaystyle{y^{\prime}_{\nu,1}}$ $\displaystyle\sim\nu+1.82109\;80\nu^{\frac{1}{3}}+0.94000\;7\nu^{-\frac{1}{3}}% -0.05808\nu^{-1}-0.0540\nu^{-\frac{5}{3}}+\dotsi.$ $\displaystyle J_{\nu}\left(j_{\nu,1}'\right)$ $\displaystyle\sim 0.67488\;51\nu^{-\frac{1}{3}}(1-0.16172\;3\nu^{-\frac{2}{3}}% +0.02918\nu^{-\frac{4}{3}}-0.0068\nu^{-2}+\cdots),$ $\displaystyle Y_{\nu}\left(y_{\nu,1}'\right)$ $\displaystyle\sim 0.57319\;40\nu^{-\frac{1}{3}}(1-0.36422\;0\nu^{-\frac{2}{3}}% +0.09077\nu^{-\frac{4}{3}}+0.0237\nu^{-2}+\cdots).$

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 $\nu\to\infty$ the following four approximations hold uniformly for $m=1,2,\dotsc$:

 10.21.41 $j_{\nu,m}=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}B_{0}(\zeta)}{2\nu}+O\left(% \frac{1}{\nu^{3}}\right),$ $\zeta=\nu^{-\frac{2}{3}}a_{m}$,
 10.21.42 $J_{\nu}'\left(j_{\nu,m}\right)=-\frac{2}{\nu^{\frac{2}{3}}}\frac{\mathrm{Ai}'% \left(a_{m}\right)}{z(\zeta)h(\zeta)}\left(1+O\left(\frac{1}{\nu^{2}}\right)% \right),$ $\zeta=\nu^{-\frac{2}{3}}a_{m}$,
 10.21.43 $j_{\nu,m}'=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}C_{0}(\zeta)}{2\zeta\nu}+O% \left(\frac{1}{\nu}\right),$ $\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}$,
 10.21.44 $J_{\nu}\left(j_{\nu,m}'\right)=\frac{h(\zeta)\mathrm{Ai}\left(a^{\prime}_{m}% \right)}{\nu^{\frac{1}{3}}}\left(1+O\left(\frac{1}{\nu^{\frac{4}{3}}}\right)% \right),$ $\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}$.

Here $a_{m}$ and $a^{\prime}_{m}$ denote respectively the zeros of the Airy function $\mathrm{Ai}\left(z\right)$ and its derivative $\mathrm{Ai}'\left(z\right)$; see §9.9. Next, $z(\zeta)$ is the inverse of the function $\zeta=\zeta(z)$ defined by (10.20.3). $B_{0}(\zeta)$ and $C_{0}(\zeta)$ are defined by (10.20.11) and (10.20.12) with $k=0$. Lastly,

 10.21.45 $h(\zeta)=\left(4\zeta/(1-z^{2})\right)^{\frac{1}{4}}.$ ⓘ Defines: $h(\zeta)$: function (locally) Symbols: $z(\zeta)$: inverse of $\zeta(z)$ A&S Ref: 9.5.36 Permalink: http://dlmf.nist.gov/10.21.E45 Encodings: TeX, pMML, png See also: Annotations for §10.21(viii), §10.21 and Ch.10

(Note: If the term $z(\zeta)(h(\zeta))^{2}C_{0}(\zeta)/(2\zeta\nu)$ in (10.21.43) is omitted, then the uniform character of the error term $O(\ifrac{1}{\nu})$ is destroyed.)

Corresponding uniform approximations for $y_{\nu,m}$, $Y_{\nu}'\left(y_{\nu,m}\right)$, $y_{\nu,m}'$, and $Y_{\nu}\left({y^{\prime}_{\nu,m}}\right)$, are obtained from (10.21.41)–(10.21.44) by changing the symbols $j$, $J$, $\mathrm{Ai}$, $\mathrm{Ai}'$, $a_{m}$, and $a^{\prime}_{m}$ to $y$, $Y$, $-\mathrm{Bi}$, $-\mathrm{Bi}'$, $b_{m}$, and $b^{\prime}_{m}$, 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 $\mathbb{C}$ 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 $Y_{n}\left(nz\right)$ and $Y_{n}'\left(nz\right)$

In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points $\pm 1$ are the boundaries of $\mathbf{K}$, that is, the eye-shaped domain depicted in Figure 10.20.3. These curves therefore intersect the imaginary axis at the points $z=\pm\mathrm{i}c$, where $c=0.66274\dotsc$.

The first set of zeros of the principal value of $Y_{n}\left(nz\right)$ are the points $z=y_{n,m}/n$, $m=1,2,\dotsc$, on the positive real axis (§10.21(i)). Secondly, there is a conjugate pair of infinite strings of zeros with asymptotes $\Im z=\pm\mathrm{i}a/n$, where

 10.21.46 $a=\tfrac{1}{2}\ln 3=0.54931\dotsc.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/10.21.E46 Encodings: TeX, pMML, png See also: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Lastly, there are two conjugate sets, with $n$ zeros in each set, that are asymptotically close to the boundary of $\mathbf{K}$ as $n\to\infty$. 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 $Y_{n}'\left(nz\right)$ have a similar pattern to those of $Y_{n}\left(nz\right)$. Figure 10.21.1: Zeros ∙⁣∙⁣∙ of Yn⁡(n⁢z) in |ph⁡z|≤π. Case n=1, -1.6≤ℜ⁡z≤2.6. Magnify Figure 10.21.3: Zeros ∙⁣∙⁣∙ of Yn⁡(n⁢z) in |ph⁡z|≤π. Case n=5, -2.6≤ℜ⁡z≤1.6. Magnify Figure 10.21.5: Zeros ∙⁣∙⁣∙ of Yn⁡(n⁢z) in |ph⁡z|≤π. Case n=10, -2.3≤ℜ⁡z≤1.9. Magnify

Zeros of ${H^{(1)}_{n}}\left(nz\right)$, ${H^{(2)}_{n}}\left(nz\right)$, ${H^{(1)}_{n}}'\left(nz\right)$, ${H^{(2)}_{n}}'\left(nz\right)$

In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points $\pm 1$ is the lower boundary of $\mathbf{K}$.

The first set of zeros of the principal value of ${H^{(1)}_{n}}\left(nz\right)$ is an infinite string with asymptote $\Im z=-\mathrm{i}d/n$, where

 10.21.47 $d=\tfrac{1}{2}\ln 2=0.34657\dotsc.$ ⓘ Defines: $d$ (locally) Symbols: $\ln\NVar{z}$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/10.21.E47 Encodings: TeX, pMML, png See also: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

The only other set comprises $n$ zeros that are asymptotically close to the lower boundary of $\mathbf{K}$ as $n\to\infty$. 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 ${H^{(1)}_{n}}'\left(nz\right)$ have a similar pattern to those of ${H^{(1)}_{n}}\left(nz\right)$. The zeros of ${H^{(2)}_{n}}\left(nz\right)$ and ${H^{(2)}_{n}}'\left(nz\right)$ are the complex conjugates of the zeros of ${H^{(1)}_{n}}\left(nz\right)$ and ${H^{(1)}_{n}}'\left(nz\right)$, respectively.

Zeros of $J_{0}\left(z\right)-\mathrm{i}J_{1}\left(z\right)$ and $J_{n}\left(z\right)-\mathrm{i}J_{n+1}\left(z\right)$

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

§10.21(x) Cross-Products

Throughout this subsection we assume $\nu\geq 0$, $x>0$, $\lambda>1$, and we denote $4\nu^{2}$ by $\mu$.

The zeros of the functions

 10.21.48 $J_{\nu}\left(x\right)Y_{\nu}\left(\lambda x\right)-Y_{\nu}\left(x\right)J_{\nu% }\left(\lambda x\right)$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.5.27 Referenced by: §10.21(x) Permalink: http://dlmf.nist.gov/10.21.E48 Encodings: TeX, pMML, png See also: Annotations for §10.21(x), §10.21 and Ch.10

and

 10.21.49 $J_{\nu}'\left(x\right)Y_{\nu}'\left(\lambda x\right)-Y_{\nu}'\left(x\right)J_{% \nu}'\left(\lambda x\right)$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.5.30 Referenced by: §10.21(x) Permalink: http://dlmf.nist.gov/10.21.E49 Encodings: TeX, pMML, png See also: Annotations for §10.21(x), §10.21 and Ch.10

are simple and the asymptotic expansion of the $m$th positive zero as $m\to\infty$ is given by

 10.21.50 $\alpha+\frac{p}{\alpha}+\frac{q-p^{2}}{\alpha^{3}}+\frac{r-4pq+2p^{3}}{\alpha^% {5}}+\dotsb,$ ⓘ Symbols: $\alpha$, $p$, $q$ and $r$ A&S Ref: 9.5.28 Referenced by: §10.21(x) Permalink: http://dlmf.nist.gov/10.21.E50 Encodings: TeX, pMML, png See also: Annotations for §10.21(x), §10.21 and Ch.10

where, in the case of (10.21.48),

 10.21.51 $\displaystyle\alpha$ $\displaystyle=\frac{m\pi}{\lambda-1},$ $\displaystyle p$ $\displaystyle=\frac{\mu-1}{8\lambda},$ $\displaystyle q$ $\displaystyle=\frac{(\mu-1)(\mu-25)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda-1)},$ $\displaystyle r$ $\displaystyle=\frac{(\mu-1)(\mu^{2}-114\mu+1073)(\lambda^{5}-1)}{5(4\lambda)^{% 5}(\lambda-1)},$ ⓘ Defines: $\alpha$ (locally), $p$ (locally), $q$ (locally) and $r$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $m$: integer A&S Ref: 9.5.29 Permalink: http://dlmf.nist.gov/10.21.E51 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.21(x), §10.21 and Ch.10

and, in the case of (10.21.49),

 10.21.52 $\displaystyle\alpha$ $\displaystyle=\frac{(m-1)\pi}{\lambda-1},$ $\displaystyle p$ $\displaystyle=\frac{\mu+3}{8\lambda},$ $\displaystyle q$ $\displaystyle=\frac{(\mu^{2}+46\mu-63)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda% -1)},$ $\displaystyle r$ $\displaystyle=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)(\lambda^{5}-1)}{5(4% \lambda)^{5}(\lambda-1)}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $\alpha$, $p$, $q$ and $r$ A&S Ref: 9.5.31 Permalink: http://dlmf.nist.gov/10.21.E52 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.21(x), §10.21 and Ch.10

The asymptotic expansion of the large positive zeros (not necessarily the $m$th) of the function

 10.21.53 $J_{\nu}'\left(x\right)Y_{\nu}\left(\lambda x\right)-Y_{\nu}'\left(x\right)J_{% \nu}\left(\lambda x\right)$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.5.32 Permalink: http://dlmf.nist.gov/10.21.E53 Encodings: TeX, pMML, png See also: Annotations for §10.21(x), §10.21 and Ch.10

is given by (10.21.50), where

 10.21.54 $\displaystyle\alpha$ $\displaystyle=\frac{(m-\tfrac{1}{2})\pi}{\lambda-1},$ $\displaystyle p$ $\displaystyle=\frac{(\mu+3)\lambda-(\mu-1)}{8\lambda(\lambda-1)},$ $\displaystyle q$ $\displaystyle=\frac{(\mu^{2}+46\mu-63)\lambda^{3}-(\mu-1)(\mu-25)}{6(4\lambda)% ^{3}(\lambda-1)},$ $\displaystyle r$ $\displaystyle=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{% 2}-114\mu+1073)}{5(4\lambda)^{5}(\lambda-1)}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $\alpha$, $p$, $q$ and $r$ A&S Ref: 9.5.33 Permalink: http://dlmf.nist.gov/10.21.E54 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.21(x), §10.21 and Ch.10

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 $(\tfrac{1}{2}\pi x)^{\frac{1}{2}}J_{\nu}\left(x\right)$ and $(\tfrac{1}{2}\pi x)^{\frac{1}{2}}Y_{\nu}\left(x\right)$. Except possibly for $x=0$ their zeros are the same as those of $J_{\nu}\left(x\right)$ and $Y_{\nu}\left(x\right)$, 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 $\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)$

For properties of the positive zeros of the function $\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)$, with $\alpha$ and $\nu$ real, see Landau (1999).

§10.21(xiii) Rayleigh Function

The Rayleigh function $\sigma_{n}\left(\nu\right)$ is defined by

 10.21.55 $\sigma_{n}\left(\nu\right)=\sum_{m=1}^{\infty}(j_{\nu,m})^{-2n},$ $n=1,2,3,\dots$. ⓘ Defines: $\sigma_{\NVar{n}}\left(\NVar{\nu}\right)$: Rayleigh function Symbols: $m$: integer, $n$: integer and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.21.E55 Encodings: TeX, pMML, png See also: Annotations for §10.21(xiii), §10.21 and Ch.10

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) $\nu$-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).