10 Bessel Functions Bessel and Hankel Functions 10.20 Uniform Asymptotic Expansions for Large Order 10.22 Integrals

§10.21 Zeros

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Referenced by:: Ch.10, §4.46
Permalink:: http://dlmf.nist.gov/10.21
See also:: Annotations for Ch.10

§10.21(i) Distribution
§10.21(ii) Analytic Properties
§10.21(iii) Infinite Products
§10.21(iv) Monotonicity Properties
§10.21(v) Inequalities
§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros
§10.21(vii) Asymptotic Expansions for Large Order
§10.21(viii) Uniform Asymptotic Approximations for Large Order
§10.21(ix) Complex Zeros
§10.21(x) Cross-Products
§10.21(xi) Riccati–Bessel Functions
§10.21(xii) Zeros of $\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)$
§10.21(xiii) Rayleigh Function
§10.21(xiv) $\nu$ -Zeros

§10.21(i) Distribution

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Defines:: $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ and ${y^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $Y_{\nu}'\left(x\right)$
Keywords:: Bessel functions, common, complex, contiguous, distribution, double, interlacing, notation, purely imaginary, zeros of Bessel functions (including derivatives), zeros of cylinder functions (including derivatives)
A&S Ref:: §9.5 (Statements re the zeros of $J_{\nu}'\left(z\right)$ and $Y_{\nu}'\left(z\right)$ in the first sentence of §9.5 are corrected here.)
Notes:: See Watson (1944, pp. 477–487), Olver (1997b, pp. 244–249), Döring (1971), and Kerimov and Skorokhodov (1985a).
Referenced by:: §10.18(iii), §10.21(ix), §10.22(ii), §10.23(iii), §10.58, §13.22, §13.9(i), §18.16(ii), §28.9, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/10.21.i
Addition (effective with 1.0.5):: The reference to Pálmai and Apagyi (2011) has been added near the middle of this subsection.
See also:: Annotations for §10.21 and Ch.10

The zeros of any cylinder function or its derivative are simple, with the possible exceptions of 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 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 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

${j^{\prime}_{0,1}}=0,$

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

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Symbols:: $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ and : integer
A&S Ref:: 9.5.1
Permalink:: http://dlmf.nist.gov/10.21.E1
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(i), §10.21 and Ch.10

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

10.21.2

$j_{\nu,1}<j_{\nu+1,1}<j_{\nu,2}<j_{\nu+1,2}<j_{\nu,3}<\dotsb,$

$y_{\nu,1}<y_{\nu+1,1}<y_{\nu,2}<y_{\nu+1,2}<y_{\nu,3}<\dotsb,$

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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

10.21.3 $\nu\leq{j^{\prime}_{\nu,1}}<y_{\nu,1}<{y^{\prime}_{\nu,1}}<j_{\nu,1}<{j^{% \prime}_{\nu,2}}<y_{\nu,2}<\dotsi.$

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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 $Y_{\nu}\left(x\right)$ , ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ , ${y^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $Y_{\nu}'\left(x\right)$ and $\nu$ : complex parameter
A&S Ref:: 9.5.2
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.21.E3
Encodings:: TeX, pMML, 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 ; see Watson (1944, §15.28).

§10.21(ii) Analytic Properties

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Keywords:: Bessel functions, analytic properties, forward differences, modulus and phase functions, relation to inverse phase functions, relation to zeros, zeros of Bessel functions (including derivatives), zeros of cylinder functions (including derivatives)
Notes:: See Watson (1944, pp. 508 and 510) and Olver (1950). (In the latter reference in (10.21.4) is replaced by .) (10.21.5) and (10.21.6) follow from (10.6.2). (10.21.12) and (10.21.13) follow from (10.18.3), (10.21.2), (10.21.3), and the fact that $\theta_{\nu}\left(x\right)$ is increasing when , whereas $\phi_{\nu}\left(x\right)$ is decreasing when $0<x<\nu$ and increasing when $x>\nu$ ; compare (10.18.8).
Referenced by:: §10.21(vii), Erratum (V1.0.1) for References
Permalink:: http://dlmf.nist.gov/10.21.ii
Addition (effective with 1.0.1):: A paragraph with an additional reference was inserted at the end of this subsection.
See also:: Annotations for §10.21 and Ch.10

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

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

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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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.5.3
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.21.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

where is a parameter, then

10.21.5 $\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)=\mathscr{C}_{\nu-1}\left(\rho_{\nu}% \right)=-\mathscr{C}_{\nu+1}\left(\rho_{\nu}\right).$

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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 $\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}% _{\nu-1}\left(\sigma_{\nu}\right)=\frac{\sigma_{\nu}}{\nu}\mathscr{C}_{\nu+1}% \left(\sigma_{\nu}\right).$

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

10.21.7

$\rho_{\nu}(0)=0,$

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

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Symbols:: ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ , $\nu$ : complex parameter, $\rho_{\nu}(t)$ : zero of cylinder function and $\sigma_{\nu}(t)$ : zero of cylinder function
A&S Ref:: 9.5.6
Permalink:: http://dlmf.nist.gov/10.21.E7
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

then

10.21.8

$j_{\nu,m}=\rho_{\nu}(m),$

$y_{\nu,m}=\rho_{\nu}(m-\tfrac{1}{2})$ , $m=1,2,\dotsc$ ,

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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 $Y_{\nu}\left(x\right)$ , : integer, $\nu$ : complex parameter and $\rho_{\nu}(t)$ : zero of cylinder function
A&S Ref:: 9.5.7
Permalink:: http://dlmf.nist.gov/10.21.E8
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

10.21.9

${j^{\prime}_{\nu,m}}=\sigma_{\nu}(m-1),$

${y^{\prime}_{\nu,m}}=\sigma_{\nu}(m-\tfrac{1}{2})$ , $m=1,2,\dotsc$ .

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Symbols:: ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ , ${y^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $Y_{\nu}'\left(x\right)$ , : integer, $\nu$ : complex parameter and $\sigma_{\nu}(t)$ : zero of cylinder function
A&S Ref:: 9.5.8
Permalink:: http://dlmf.nist.gov/10.21.E9
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

10.21.10

$\mathscr{C}_{\nu}'\left(\rho_{\nu}\right)=\left(\frac{\rho_{\nu}}{2}\frac{% \mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{-\frac{1}{2}},$

$\mathscr{C}_{\nu}\left(\sigma_{\nu}\right)=\left(\frac{\sigma_{\nu}^{2}-\nu^{2% }}{2\sigma_{\nu}}\frac{\mathrm{d}\sigma_{\nu}}{\mathrm{d}t}\right)^{-\frac{1}{% 2}},$

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Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\nu$ : complex parameter, $\rho_{\nu}(t)$ : zero of cylinder function and $\sigma_{\nu}(t)$ : zero of cylinder function
A&S Ref:: 9.5.9
Permalink:: http://dlmf.nist.gov/10.21.E10
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

10.21.11 $2\rho_{\nu}^{2}\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\frac{{\mathrm{d}}^{3}% \rho_{\nu}}{{\mathrm{d}t}^{3}}-3\rho_{\nu}^{2}\*\left(\frac{{\mathrm{d}}^{2}% \rho_{\nu}}{{\mathrm{d}t}^{2}}\right)^{2}-4\pi^{2}\rho_{\nu}^{2}\*\left(\frac{% \mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{2}+(4\rho_{\nu}^{2}+1-4\nu^{2})% \left(\frac{\mathrm{d}\rho_{\nu}}{\mathrm{d}t}\right)^{4}=0.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\nu$ : complex parameter and $\rho_{\nu}(t)$ : zero of cylinder function
Permalink:: http://dlmf.nist.gov/10.21.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

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

$\theta_{\nu}\left(j_{\nu,m}\right)=(m-\tfrac{1}{2})\pi,$

$\theta_{\nu}\left(y_{\nu,m}\right)=(m-1)\pi,$ $m=1,2,\dotsc$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer and $\nu$ : complex parameter
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.21.E12
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

10.21.13

$\phi_{\nu}\left({j^{\prime}_{\nu,m}}\right)=(m-\tfrac{1}{2})\pi,$

$\phi_{\nu}\left({y^{\prime}_{\nu,m}}\right)=m\pi,$ $m=1,2,\dotsc$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of derivatives of Bessel functions, ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ , ${y^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $Y_{\nu}'\left(x\right)$ , : integer and $\nu$ : complex parameter
Referenced by:: §10.21(ii)
Permalink:: http://dlmf.nist.gov/10.21.E13
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.21(ii), §10.21 and Ch.10

For sign properties of the forward differences that are defined by

10.21.14

$\Delta\rho_{\nu}(t)=\rho_{\nu}(t+1)-\rho_{\nu}(t),$

$\Delta^{2}\rho_{\nu}(t)=\Delta\rho_{\nu}(t+1)-\Delta\rho_{\nu}(t),\dotsc,$

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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 is given in Muldoon and Spigler (1984).

§10.21(iii) Infinite Products

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Keywords:: Bessel functions, infinite products
Notes:: For (10.21.15) see Watson (1944, pp. 497–498). (10.21.16) is proved by a similar method.
Referenced by:: §10.58
Permalink:: http://dlmf.nist.gov/10.21.iii
See also:: Annotations for §10.21 and Ch.10

10.21.15 $J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}% \prod_{k=1}^{\infty}\left(1-\frac{z^{2}}{{j_{\nu,k}}^{2}}\right),$ $\nu\geq 0$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.5.10
Referenced by:: §10.21(iii)
Permalink:: http://dlmf.nist.gov/10.21.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(iii), §10.21 and Ch.10

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${j^{\prime}_{\NVar{\nu},\NVar{m}}}$ : zeros of the Bessel function derivative $J_{\nu}'\left(x\right)$ , : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.5.11
Referenced by:: §10.21(iii)
Permalink:: http://dlmf.nist.gov/10.21.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(iii), §10.21 and Ch.10

§10.21(iv) Monotonicity Properties

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Keywords:: monotonicity, zeros of Bessel functions (including derivatives), zeros of cylinder functions (including derivatives)
Notes:: See Watson (1944, pp. 508–510), Lorch (1990, 1995), Wong and Lang (1991), McCann (1977), Lewis and Muldoon (1977), and Mercer (1992).
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/10.21.iv
Addition (effective with 1.0.5):: The reference to Lorch and Muldoon (2008) has been added at the end of this subsection.
See also:: Annotations for §10.21 and Ch.10

Any positive zero 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 $\frac{\mathrm{d}c}{\mathrm{d}\nu}=2c\int_{0}^{\infty}K_{0}\left(2c\sinh t% \right)e^{-2\nu t}\,\mathrm{d}t,$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\nu$ : complex parameter and : zero of cylinder function
Permalink:: http://dlmf.nist.gov/10.21.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(iv), §10.21 and Ch.10

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

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\nu$ : complex parameter and : zero of cylinder function
Permalink:: http://dlmf.nist.gov/10.21.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(iv), §10.21 and Ch.10

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

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Keywords:: bounds, purely imaginary, zeros of Bessel functions (including derivatives)
Permalink:: http://dlmf.nist.gov/10.21.v
See also:: Annotations for §10.21 and Ch.10

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

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Keywords:: McMahon’s asymptotic expansions, asymptotic expansions for large zeros, error bounds, zeros of Bessel functions, zeros of cylinder functions (including derivatives)
Notes:: For (10.21.19) see Watson (1944, pp. 503–507) or Olver (1997b, pp. 247–248). Similar methods can be used for (10.21.20).
Referenced by:: §10.21(xi), §10.58, §10.74(vi), §13.9(i), §13.9(ii)
Permalink:: http://dlmf.nist.gov/10.21.vi
See also:: Annotations for §10.21 and Ch.10

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

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

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer and $\nu$ : complex parameter
A&S Ref:: 9.5.12
Referenced by:: §10.21(vi), §10.21(vi)
Permalink:: http://dlmf.nist.gov/10.21.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vi), §10.21 and Ch.10

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 .

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

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer and $\nu$ : complex parameter
A&S Ref:: 9.5.13
Referenced by:: §10.21(vi), §10.21(vi)
Permalink:: http://dlmf.nist.gov/10.21.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vi), §10.21 and Ch.10

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 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, : integer, $\nu$ : complex parameter and : 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

ⓘ

Keywords:: asymptotic expansions for large order, uniform, zeros of cylinder functions (including derivatives)
Notes:: See Olver (1951, 1952).
Referenced by:: §10.21(xi)
Permalink:: http://dlmf.nist.gov/10.21.vii
See also:: Annotations for §10.21 and Ch.10

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 fixed,

10.21.22 $\rho_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, : nonnegative integer, $\nu$ : complex parameter, $\rho_{\nu}(t)$ : zero of cylinder function and $\alpha$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $M\left(\NVar{z}\right)$ : Airy modulus function, : nonnegative integer, $\nu$ : complex parameter, $\rho_{\nu}(t)$ : zero of cylinder function, $\alpha$ : coefficients and $\beta$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

$\alpha_{0}=1,$

$\alpha_{1}=\alpha,$

$\alpha_{2}=\tfrac{3}{10}\alpha^{2},$

$\alpha_{3}=-\tfrac{1}{350}\alpha^{3}+\tfrac{1}{70},$

$\alpha_{4}=-\tfrac{479}{63000}\alpha^{4}-\tfrac{1}{3150}\alpha,$

$\alpha_{5}=\tfrac{20231}{80\;85000}\alpha^{5}-\tfrac{551}{1\;61700}\alpha^{2},$

ⓘ

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

$\beta_{0}=1,$

$\beta_{1}=-\tfrac{4}{5}\alpha,$

$\beta_{2}=\tfrac{18}{35}\alpha^{2},$

$\beta_{3}=-\tfrac{88}{315}\alpha^{3}-\tfrac{11}{1575},$

$\beta_{4}=\tfrac{79586}{6\;06375}\alpha^{4}+\tfrac{9824}{6\;06375}\alpha.$

ⓘ

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 $(>-\tfrac{1}{6})$ fixed,

10.21.27 $\sigma_{\nu}(t)\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, : nonnegative integer, $\nu$ : complex parameter, $\sigma_{\nu}(t)$ : zero of cylinder function and $\alpha^{\prime}$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $N\left(\NVar{z}\right)$ : Airy modulus function, : nonnegative integer, $\nu$ : complex parameter, $\sigma_{\nu}(t)$ : zero of cylinder function, $\beta$ : coefficients and $\alpha^{\prime}$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

$\alpha^{\prime}_{0}=1,$

$\alpha^{\prime}_{1}=\alpha^{\prime},$

$\alpha^{\prime}_{2}=\tfrac{3}{10}{\alpha^{\prime}}^{2}-\tfrac{1}{10}{\alpha^{% \prime}}^{-1},$

$\alpha^{\prime}_{3}=-\tfrac{1}{350}{\alpha^{\prime}}^{3}-\tfrac{1}{25}-\tfrac{% 1}{200}{\alpha^{\prime}}^{-3},$

$\alpha^{\prime}_{4}=-\tfrac{479}{63000}{\alpha^{\prime}}^{4}+\tfrac{509}{31500% }\alpha^{\prime}+\tfrac{1}{1500}{\alpha^{\prime}}^{-2}-\tfrac{1}{2000}{\alpha^% {\prime}}^{-5},$

ⓘ

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

$\beta^{\prime}_{0}=1,$

$\beta^{\prime}_{1}=-\tfrac{1}{5}\alpha^{\prime},$

$\beta^{\prime}_{2}=\tfrac{9}{350}{\alpha^{\prime}}^{2}+\tfrac{1}{100}{\alpha^{% \prime}}^{-1},$

$\beta^{\prime}_{3}=\tfrac{89}{15750}{\alpha^{\prime}}^{3}-\tfrac{47}{4500}+% \tfrac{1}{3000}{\alpha^{\prime}}^{-3}.$

ⓘ

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 $j_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\alpha$ : coefficients
Referenced by:: §10.21(vii), §10.21(vii)
Permalink:: http://dlmf.nist.gov/10.21.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

10.21.33 $y_{\nu,m}\sim\nu\sum_{k=0}^{\infty}\frac{\alpha_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\alpha$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$ : Airy modulus function, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\beta$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $M\left(\NVar{z}\right)$ : Airy modulus function, $b_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\beta$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

and

10.21.36 $j_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\alpha^{\prime}$ : coefficients
Referenced by:: §10.21(vii)
Permalink:: http://dlmf.nist.gov/10.21.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

10.21.37 $y_{\nu,m}'\sim\nu\sum_{k=0}^{\infty}\frac{\alpha^{\prime}_{k}}{\nu^{2k/3}},$

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\alpha^{\prime}$ : coefficients
Referenced by:: §10.21(vii)
Permalink:: http://dlmf.nist.gov/10.21.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $N\left(\NVar{z}\right)$ : Airy modulus function, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\beta$ : coefficients
Permalink:: http://dlmf.nist.gov/10.21.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $N\left(\NVar{z}\right)$ : Airy modulus function, $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , $b^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Bi}'$ , : integer, : nonnegative integer, $\nu$ : complex parameter and $\beta$ : coefficients
Referenced by:: §10.21(vii)
Permalink:: http://dlmf.nist.gov/10.21.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(vii), §10.21 and Ch.10

Here $a_{m}$ , $b_{m}$ , $a^{\prime}_{m}$ , $b^{\prime}_{m}$ are the th negative zeros of $\operatorname{Ai}\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ , 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

$j_{\nu,1}\sim\nu+1.85575\;71\nu^{\frac{1}{3}}+1.03315\;0\nu^{-\frac{1}{3}}-0.0% 0397\nu^{-1}-0.0908\nu^{-\frac{5}{3}}+0.043\nu^{-\frac{7}{3}}+\dotsb,$

$y_{\nu,1}\sim\nu+0.93157\;68\nu^{\frac{1}{3}}+0.26035\;1\nu^{-\frac{1}{3}}+0.0% 1198\nu^{-1}-0.0060\nu^{-\frac{5}{3}}-0.001\nu^{-\frac{7}{3}}+\dotsb,$

$J_{\nu}'\left(j_{\nu,1}\right)\sim-1.11310\;28\nu^{-\frac{2}{3}}\div(1+1.48460% \;6\nu^{-\frac{2}{3}}+0.43294\nu^{-\frac{4}{3}}-0.1943\nu^{-2}+0.019\nu^{-% \frac{8}{3}}+\cdots),$

$Y_{\nu}'\left(y_{\nu,1}\right)\sim 0.95554\;86\nu^{-\frac{2}{3}}\div(1+0.74526% \;1\nu^{-\frac{2}{3}}+0.10910\nu^{-\frac{4}{3}}-0.0185\nu^{-2}-0.003\nu^{-% \frac{8}{3}}+\cdots),$

${j^{\prime}_{\nu,1}}\sim\nu+0.80861\;65\nu^{\frac{1}{3}}+0.07249\;0\nu^{-\frac% {1}{3}}-0.05097\nu^{-1}+0.0094\nu^{-\frac{5}{3}}+\dotsb,$

${y^{\prime}_{\nu,1}}\sim\nu+1.82109\;80\nu^{\frac{1}{3}}+0.94000\;7\nu^{-\frac% {1}{3}}-0.05808\nu^{-1}-0.0540\nu^{-\frac{5}{3}}+\dotsi.$

$J_{\nu}\left(j_{\nu,1}'\right)\sim 0.67488\;51\nu^{-\frac{1}{3}}(1-0.16172\;3% \nu^{-\frac{2}{3}}+0.02918\nu^{-\frac{4}{3}}-0.0068\nu^{-2}+\cdots),$

$Y_{\nu}\left(y_{\nu,1}'\right)\sim 0.57319\;40\nu^{-\frac{1}{3}}(1-0.36422\;0% \nu^{-\frac{2}{3}}+0.09077\nu^{-\frac{4}{3}}+0.0237\nu^{-2}+\cdots).$

For numerical coefficients for see Olver (1951, Tables 3–6).

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

§10.21(viii) Uniform Asymptotic Approximations for Large Order

ⓘ

Keywords:: asymptotic expansions for large order, uniform, zeros of Bessel functions (including derivatives)
Referenced by:: §10.21(vii), §10.58, 2nd item
Permalink:: http://dlmf.nist.gov/10.21.viii
See also:: Annotations for §10.21 and Ch.10

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

10.21.41 $j_{\nu,m}=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}B_{0}(\zeta)}{2\nu}+O\left(% \frac{1}{\nu^{3}}\right),$ $\zeta=\nu^{-\frac{2}{3}}a_{m}$ ,

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : integer, $\nu$ : complex parameter, $B_{k}(\zeta)$ : coefficients, $z(\zeta)$ : inverse of $\zeta(z)$ and $h(\zeta)$ : function
A&S Ref:: 9.5.22
Referenced by:: §10.21(viii)
Permalink:: http://dlmf.nist.gov/10.21.E41
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(viii), §10.21 and Ch.10

10.21.42 $J_{\nu}'\left(j_{\nu,m}\right)=-\frac{2}{\nu^{\frac{2}{3}}}\frac{\operatorname% {Ai}'\left(a_{m}\right)}{z(\zeta)h(\zeta)}\left(1+O\left(\frac{1}{\nu^{2}}% \right)\right),$ $\zeta=\nu^{-\frac{2}{3}}a_{m}$ ,

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}$ , $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , : integer, $\nu$ : complex parameter, $z(\zeta)$ : inverse of $\zeta(z)$ and $h(\zeta)$ : function
A&S Ref:: 9.5.23
Permalink:: http://dlmf.nist.gov/10.21.E42
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(viii), §10.21 and Ch.10

10.21.43 $j_{\nu,m}'=\nu z(\zeta)+\frac{z(\zeta)(h(\zeta))^{2}C_{0}(\zeta)}{2\zeta\nu}+O% \left(\frac{1}{\nu}\right),$ $\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}$ ,

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , : integer, $\nu$ : complex parameter, $C_{k}(\zeta)$ : coefficients, $z(\zeta)$ : inverse of $\zeta(z)$ and $h(\zeta)$ : function
A&S Ref:: 9.5.24
Referenced by:: §10.21(viii)
Permalink:: http://dlmf.nist.gov/10.21.E43
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(viii), §10.21 and Ch.10

10.21.44 $J_{\nu}\left(j_{\nu,m}'\right)=\frac{h(\zeta)\operatorname{Ai}\left(a^{\prime}% _{m}\right)}{\nu^{\frac{1}{3}}}\left(1+O\left(\frac{1}{\nu^{\frac{4}{3}}}% \right)\right),$ $\zeta=\nu^{-\frac{2}{3}}a^{\prime}_{m}$ .

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $a^{\prime}_{\NVar{k}}$ : th zero of Airy $\operatorname{Ai}'$ , : integer, $\nu$ : complex parameter and $h(\zeta)$ : function
A&S Ref:: 9.5.25
Referenced by:: §10.21(viii)
Permalink:: http://dlmf.nist.gov/10.21.E44
Encodings:: TeX, pMML, png
See also:: Annotations for §10.21(viii), §10.21 and Ch.10

Here $a_{m}$ and $a^{\prime}_{m}$ denote respectively the zeros of the Airy function $\operatorname{Ai}\left(z\right)$ and its derivative $\operatorname{Ai}'\left(z\right)$ ; 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 . 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 , , $\operatorname{Ai}$ , $\operatorname{Ai}'$ , $a_{m}$ , and $a^{\prime}_{m}$ to , , $-\operatorname{Bi}$ , $-\operatorname{Bi}'$ , $b_{m}$ , 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

ⓘ

Keywords:: Hankel functions, complex, distribution, zeros, zeros of Bessel functions (including derivatives)
Notes:: The zeros depicted in Figures 10.21.1–10.21.6 were computed at NIST using methods referred to in §10.74(vi).
Referenced by:: §10.74(vi), Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/10.21.ix
Addition (effective with 1.0.10):: A sentence and reference to Segura (2013) were added at the end of the second paragraph of this subsection.
See also:: Annotations for §10.21 and Ch.10

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 . 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).

Zeros of $Y_{n}\left(nz\right)$ and $Y_{n}'\left(nz\right)$

ⓘ

See also:: Annotations for §10.21(ix), §10.21 and Ch.10

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 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 , 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 $\bullet\bullet\bullet$ of $Y_{n}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-1.6\leq\Re z\leq 2.6$ . Magnify

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix), §10.21(ix)
Permalink:: http://dlmf.nist.gov/10.21.F1
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Figure 10.21.2: Zeros $\bullet\bullet\bullet$ of ${H^{(1)}_{n}}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-2.8\leq\Re z\leq 1.4$ . Magnify

ⓘ

Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix), §10.42
Permalink:: http://dlmf.nist.gov/10.21.F2
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Figure 10.21.3: Zeros $\bullet\bullet\bullet$ of $Y_{n}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-2.6\leq\Re z\leq 1.6$ . Magnify

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix), §10.21(ix)
Permalink:: http://dlmf.nist.gov/10.21.F3
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Figure 10.21.4: Zeros $\bullet\bullet\bullet$ of ${H^{(1)}_{n}}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-2.6\leq\Re z\leq 1.6$ . Magnify

ⓘ

Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix), §10.42
Permalink:: http://dlmf.nist.gov/10.21.F4
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Figure 10.21.5: Zeros $\bullet\bullet\bullet$ of $Y_{n}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-2.3\leq\Re z\leq 1.9$ . Magnify

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix)
Permalink:: http://dlmf.nist.gov/10.21.F5
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

Figure 10.21.6: Zeros $\bullet\bullet\bullet$ of ${H^{(1)}_{n}}\left(nz\right)$ in $|\operatorname{ph}z|\leq\pi$ . Case

, $-2.3\leq\Re z\leq 1.9$ . Magnify

ⓘ

Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : integer and : complex variable
Referenced by:: §10.21(ix), §10.21(ix), §10.21(ix), §10.21(ix), §10.42
Permalink:: http://dlmf.nist.gov/10.21.F6
Encodings:: pdf, png
See also:: Annotations for §10.21(ix), §10.21(ix), §10.21 and Ch.10

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

ⓘ

See also:: Annotations for §10.21(ix), §10.21 and Ch.10

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:: (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 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 , 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 $J_{n}\left(z\right)-\mathrm{i}J_{n+1}\left(z\right)$

ⓘ

Keywords:: Hankel functions, complex, derivatives, zeros, zeros of Bessel functions (including derivatives)
See also:: Annotations for §10.21(ix), §10.21 and Ch.10

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

§10.21(x) Cross-Products

ⓘ

Keywords:: Bessel functions, asymptotic expansions, cross-products, of cross-products, zeros, zeros of Bessel functions (including derivatives)
Notes:: See McMahon (1894), Gray et al. (1922, p. 261), and Cochran (1964).
Referenced by:: §10.21(xi), §10.58
Permalink:: http://dlmf.nist.gov/10.21.x
See also:: Annotations for §10.21 and Ch.10

Throughout this subsection we assume $\nu\geq 0$ , , $\lambda>1$ , 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, : 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, : 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 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:: $\alpha$ , , and
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

$\alpha=\frac{m\pi}{\lambda-1},$

$p=\frac{\mu-1}{8\lambda},$

$q=\frac{(\mu-1)(\mu-25)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda-1)},$

$r=\frac{(\mu-1)(\mu^{2}-114\mu+1073)(\lambda^{5}-1)}{5(4\lambda)^{5}(\lambda-1% )},$

ⓘ

Defines:: $\alpha$ (locally), (locally), (locally) and (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter and : 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

$\alpha=\frac{(m-1)\pi}{\lambda-1},$

$p=\frac{\mu+3}{8\lambda},$

$q=\frac{(\mu^{2}+46\mu-63)(\lambda^{3}-1)}{6(4\lambda)^{3}(\lambda-1)},$

$r=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)(\lambda^{5}-1)}{5(4\lambda)^{5}(% \lambda-1)}.$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $\alpha$ , , and
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 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, : 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

$\alpha=\frac{(m-\tfrac{1}{2})\pi}{\lambda-1},$

$p=\frac{(\mu+3)\lambda-(\mu-1)}{8\lambda(\lambda-1)},$

$q=\frac{(\mu^{2}+46\mu-63)\lambda^{3}-(\mu-1)(\mu-25)}{6(4\lambda)^{3}(\lambda% -1)},$

$r=\frac{(\mu^{3}+185\mu^{2}-2053\mu+1899)\lambda^{5}-(\mu-1)(\mu^{2}-114\mu+10% 73)}{5(4\lambda)^{5}(\lambda-1)}.$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $\alpha$ , , and
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

ⓘ

Keywords:: Riccati–Bessel functions, zeros
Permalink:: http://dlmf.nist.gov/10.21.xi
See also:: Annotations for §10.21 and Ch.10

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 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 $\alpha J_{\nu}\left(x\right)+xJ_{\nu}'\left(x\right)$

ⓘ

Permalink:: http://dlmf.nist.gov/10.21.xii
See also:: Annotations for §10.21 and Ch.10

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

ⓘ

Keywords:: Rayleigh function, distribution, zeros of Bessel functions (including derivatives)
Referenced by:: §10.73(v), Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/10.21.xiii
Addition (effective with 1.0.5):: The reference to Kerimov (2008) has been added at the end of this subsection.
See also:: Annotations for §10.21 and Ch.10

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:: : integer, : 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

ⓘ

Keywords:: Hankel functions, with respect to order ( $\nu$ -zeros), zeros, zeros of Bessel functions (including derivatives)
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/10.21.xiv
Addition (effective with 1.0.5):: The reference to Sandström and Ackrén (2007) has been added at the end of this subsection.
See also:: Annotations for §10.21 and Ch.10

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).

10.20 Uniform Asymptotic Expansions for Large Order 10.22 Integrals

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