§10.21 Zeros

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 $J_{\nu }\left(z\right)$, $J_{\nu }^{\prime }\left(z\right)$, $Y_{\nu }\left(z\right)$, and $Y_{\nu }^{\prime }\left(z\right)$, each have an infinite number of positive real zeros. All of these zeros are simple, provided that $\nu \ge -1$ in the case of $J_{\nu }^{\prime }\left(z\right)$, and $\nu \ge -\frac{1}{2}$ in the case of $Y_{\nu }^{\prime }\left(z\right)$. When all of their zeros are simple, the $m$th positive zeros of these functions are denoted by $j_{\nu ,m}$, $j_{\nu ,m}^{\prime }$, $y_{\nu ,m}$, and $y_{\nu ,m}^{\prime }$ respectively, except that $z=0$ is counted as the first zero of $J_0^{\prime }\left(z\right)$. Since $J_0^{\prime }\left(z\right)=-J_1\left(z\right)$ we have

10.21.1		$j_{0,1}^{\prime }$	$=0,$
		$j_{0,m}^{\prime }$	$=j_{1,m-1},$
	$m=2,3,\mathrm{\dots }$.
ⓘ Symbols: $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$ and $m$: 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 $\nu \ge 0$, the zeros interlace according to the inequalities

10.21.2		$j_{\nu ,1}$
		$y_{\nu ,1}$
ⓘ Symbols: $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: 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
ⓘ Symbols: $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$, $y_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $Y_{\nu }^{\prime }\left(x\right)$ 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 ${𝒞}_{\nu }\left(z\right)$ and the contiguous function ${𝒞}_{\nu +1}\left(z\right)$. See also Elbert and Laforgia (1994).

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

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

For information on the real double zeros of $J_{\nu }^{\prime }\left(z\right)$ and $Y_{\nu }^{\prime }\left(z\right)$ when and , respectively, see Döring (1971) and Kerimov and Skorokhodov (1986). The latter reference also has information on double zeros of the second and third derivatives of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$.

No two of the functions $J_0\left(z\right)$, $J_1\left(z\right)$, $J_2\left(z\right),\mathrm{\dots }$, have any common zeros other than $z=0$; see Watson (1944, §15.28).

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

10.21.4		$${𝒞}_{\nu }\left(z\right)=J_{\nu }\left(z\right)\cos \left(\pi t\right)+Y_{\nu }\left(z\right)\sin \left(\pi t\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\sin z$: sine function, $z$: 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 $t$ is a parameter, then

10.21.5		$${𝒞}_{\nu }^{\prime }\left({\rho }_{\nu }\right)={𝒞}_{\nu -1}\left({\rho }_{\nu }\right)=-{𝒞}_{\nu +1}\left({\rho }_{\nu }\right).$$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: 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 ${𝒞}_{\nu }^{\prime }\left(z\right)$, then

10.21.6		$${𝒞}_{\nu }\left({\sigma }_{\nu }\right)=\frac{{\sigma }_{\nu }}{\nu }{𝒞}_{\nu -1}\left({\sigma }_{\nu }\right)=\frac{{\sigma }_{\nu }}{\nu }{𝒞}_{\nu +1}\left({\sigma }_{\nu }\right).$$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: 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 $\nu \ge 0$ and these functions are fixed by

10.21.7		${\rho }_{\nu }(0)$	$=0,$
		${\sigma }_{\nu }(0)$	$=j_{\nu ,1}^{\prime },$
ⓘ Symbols: $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$, $\nu$: 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		$j_{\nu ,m}$	$={\rho }_{\nu }(m),$
		$y_{\nu ,m}$	$={\rho }_{\nu }(m-\frac{1}{2})$,
	$m=1,2,\mathrm{\dots }$,
ⓘ Symbols: $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: 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		$j_{\nu ,m}^{\prime }$	$={\sigma }_{\nu }(m-1),$
		$y_{\nu ,m}^{\prime }$	$={\sigma }_{\nu }(m-\frac{1}{2})$,
	$m=1,2,\mathrm{\dots }$.
ⓘ Symbols: $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$, $y_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $Y_{\nu }^{\prime }\left(x\right)$, $m$: 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		${𝒞}_{\nu }^{\prime }\left({\rho }_{\nu }\right)$	$={\left({\displaystyle \frac{{\rho }_{\nu }}{2}}{\displaystyle \frac{d{\rho }_{\nu }}{dt}}\right)}^{-\frac{1}{2}},$
		${𝒞}_{\nu }\left({\sigma }_{\nu }\right)$	$={\left({\displaystyle \frac{{\sigma }_{\nu }^2-{\nu }^2}{2{\sigma }_{\nu }}}{\displaystyle \frac{d{\sigma }_{\nu }}{dt}}\right)}^{-\frac{1}{2}},$
ⓘ Symbols: ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\nu$: 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		$$2{\rho }_{\nu }^2\frac{d{\rho }_{\nu }}{dt}\frac{d^3{\rho }_{\nu }}{{dt}^3}-3{\rho }_{\nu }^2{\left(\frac{d^2{\rho }_{\nu }}{{dt}^2}\right)}^2-4{\pi }^2{\rho }_{\nu }^2{\left(\frac{d{\rho }_{\nu }}{dt}\right)}^2+(4{\rho }_{\nu }^2+1-4{\nu }^2){\left(\frac{d{\rho }_{\nu }}{dt}\right)}^4=0.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\nu$: 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 ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then

10.21.12		${\theta }_{\nu }\left(j_{\nu ,m}\right)$	$=(m-\frac{1}{2})\pi ,$
		${\theta }_{\nu }\left(y_{\nu ,m}\right)$	$=(m-1)\pi ,$
	$m=1,2,\mathrm{\dots }$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: 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		${\varphi }_{\nu }\left(j_{\nu ,m}^{\prime }\right)$	$=(m-\frac{1}{2})\pi ,$
		${\varphi }_{\nu }\left(y_{\nu ,m}^{\prime }\right)$	$=m\pi ,$
	$m=1,2,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$, $y_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $Y_{\nu }^{\prime }\left(x\right)$, $m$: 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		$\mathrm{\Delta }{\rho }_{\nu }(t)$	$={\rho }_{\nu }(t+1)-{\rho }_{\nu }(t),$
		${\mathrm{\Delta }}^2{\rho }_{\nu }(t)$	$=\mathrm{\Delta }{\rho }_{\nu }(t+1)-\mathrm{\Delta }{\rho }_{\nu }(t),\mathrm{\dots },$
ⓘ Symbols: $\mathrm{\Delta }$: 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 $t=1,2,3,\mathrm{\dots }$, 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).

10.21.15		$J_{\nu }\left(z\right)$	$={\displaystyle \frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}}{\displaystyle \prod _{k=1}^{\mathrm{\infty }}}\left(1-{\displaystyle \frac{z^2}{j_{\nu ,k}^2}}\right),$
$\nu \ge 0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $k$: nonnegative integer, $z$: 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		$J_{\nu }^{\prime }\left(z\right)$	$={\displaystyle \frac{{(\frac{1}{2}z)}^{\nu -1}}{2\mathrm{\Gamma }\left(\nu \right)}}{\displaystyle \prod _{k=1}^{\mathrm{\infty }}}\left(1-{\displaystyle \frac{z^2}{j_{\nu ,k}^{\prime }{}_{}{}^2}}\right),$
$\nu >0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$, $k$: nonnegative integer, $z$: 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

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

10.21.17		$$\frac{dc}{d\nu }=2c{\int }_0^{\mathrm{\infty }}{K}_0\left(2c\sinh t\right){\mathrm{e}}^{-2\nu t}dt,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\nu$: complex parameter and $c$: 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		$$\frac{d{c}^{\prime }}{d\nu }=\frac{2c^{\prime }}{c_{}^{\prime }{}_{}{}^2-{\nu }^2}{\int }_0^{\mathrm{\infty }}(c_{}^{\prime }{}_{}{}^2\cosh \left(2t\right)-{\nu }^2)K_0\left(2c^{\prime }\sinh t\right){\mathrm{e}}^{-2\nu t}dt,$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\nu$: complex parameter and $c$: 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, $j_{\nu ,m}$, $y_{\nu ,m}$, $j_{\nu ,m}^{\prime }$, and $y_{\nu ,m}^{\prime }$ are increasing functions of $\nu$ when $\nu \ge 0$. It is also true that the positive zeros $j_{\nu }^{\prime \prime }$ and $j_{\nu }^{\prime \prime \prime }$ of $J_{\nu }^{\prime \prime }\left(x\right)$ and $J_{\nu }^{\prime \prime \prime }\left(x\right)$, respectively, are increasing functions of $\nu$ when $\nu >0$, provided that in the latter case $j_{\nu }^{\prime \prime \prime }>\sqrt{3}$ when .

$j_{\nu ,m}/\nu$ and $j_{\nu ,m}^{\prime }/\nu$ are decreasing functions of $\nu$ when $\nu >0$ for $m=1,2,3,\mathrm{\dots }$.

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).

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).

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

10.21.19		$$j_{\nu ,m},y_{\nu ,m}\sim \begin{array}{l}a-\frac{\mu -1}{8a}-\frac{4(\mu -1)(7\mu -31)}{3{(8a)}^3}-\frac{32(\mu -1)(83{\mu }^2-982\mu +3779)}{15{(8a)}^5}\\ -\frac{64(\mu -1)(6949{\mu }^3-\mathrm{1\hspace{0.33em}53855}{\mu }^2+\mathrm{15\hspace{0.33em}85743}\mu -\mathrm{62\hspace{0.33em}77237})}{105{(8a)}^7}-\mathrm{\cdots },\end{array}$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: 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 $a=(m+\frac{1}{2}\nu -\frac{1}{4})\pi$ for $j_{\nu ,m}$, $a=(m+\frac{1}{2}\nu -\frac{3}{4})\pi$ for $y_{\nu ,m}$. With $a=(t+\frac{1}{2}\nu -\frac{1}{4})\pi$, the right-hand side is the asymptotic expansion of ${\rho }_{\nu }(t)$ for large $t$.

10.21.20		$$j_{\nu ,m}^{\prime },y_{\nu ,m}^{\prime }\sim \begin{array}{l}b-\frac{\mu +3}{8b}-\frac{4(7{\mu }^2+82\mu -9)}{3{(8b)}^3}-\frac{32(83{\mu }^3+2075{\mu }^2-3039\mu +3537)}{15{(8b)}^5}\\ -\frac{64(6949{\mu }^4+\mathrm{2\hspace{0.33em}96492}{\mu }^3-\mathrm{12\hspace{0.33em}48002}{\mu }^2+\mathrm{74\hspace{0.33em}14380}\mu -\mathrm{58\hspace{0.33em}53627})}{105{(8b)}^7}-\mathrm{\cdots },\end{array}$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: 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 $b=(m+\frac{1}{2}\nu -\frac{3}{4})\pi$ for $j_{\nu ,m}^{\prime }$, $b=(m+\frac{1}{2}\nu -\frac{1}{4})\pi$ for $y_{\nu ,m}^{\prime }$, and $b=(t+\frac{1}{2}\nu +\frac{1}{4})\pi$ 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 $j_{\nu ,m}^{\prime \prime }$ of $J_{\nu }^{\prime \prime }\left(x\right)$ Wong and Lang (1990) gives the corresponding expansion

10.21.21		$$j_{\nu ,m}^{\prime \prime }\sim c-\frac{\mu +7}{8c}-\frac{28{\mu }^2+424\mu +1724}{3{(8c)}^3}-\mathrm{\cdots },$$
ⓘ 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 $c=(m+\frac{1}{2}\nu -\frac{1}{4})\pi$ if , and $c=(m+\frac{1}{2}\nu -\frac{5}{4})\pi$ if $\nu >1$. An error bound is included for the case $\nu \ge \frac{3}{2}$.

Let ${𝒞}_{\nu }\left(x\right)$, ${\rho }_{\nu }(t)$, 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 $\varphi \left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8.

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

10.21.22		$${\rho }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $k$: 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		$${𝒞}_{\nu }^{\prime }\left({\rho }_{\nu }(t)\right)\sim \frac{{(2/\nu )}^{\frac{2}{3}}}{\pi M\left(-2^{\frac{1}{3}}\alpha \right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $M\left(z\right)$: Airy modulus function, $k$: 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 $\theta \left(z\right)$: 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		${\alpha }_0$	$=1,$
		${\alpha }_1$	$=\alpha ,$
		${\alpha }_2$	$=\frac{3}{10}{\alpha }^2,$
		${\alpha }_3$	$=-\frac{1}{350}{\alpha }^3+\frac{1}{70},$
		${\alpha }_4$	$=-\frac{479}{63000}{\alpha }^4-\frac{1}{3150}\alpha ,$
		${\alpha }_5$	$=\frac{20231}{\mathrm{80\hspace{0.33em}85000}}{\alpha }^5-\frac{551}{\mathrm{1\hspace{0.33em}61700}}{\alpha }^2,$
ⓘ 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		${\beta }_0$	$=1,$
		${\beta }_1$	$=-\frac{4}{5}\alpha ,$
		${\beta }_2$	$=\frac{18}{35}{\alpha }^2,$
		${\beta }_3$	$=-\frac{88}{315}{\alpha }^3-\frac{11}{1575},$
		${\beta }_4$	$=\frac{79586}{\mathrm{6\hspace{0.33em}06375}}{\alpha }^4+\frac{9824}{\mathrm{6\hspace{0.33em}06375}}\alpha .$
ⓘ 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 $\nu \to \mathrm{\infty }$ with $t$ $(>-\frac{1}{6})$ fixed,

10.21.27		$${\sigma }_{\nu }(t)\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $k$: 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		$${𝒞}_{\nu }\left({\sigma }_{\nu }(t)\right)\sim \frac{{(2/\nu )}^{\frac{1}{3}}}{\pi N\left(-2^{\frac{1}{3}}{\alpha }^{\prime }\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k^{\prime }}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $N\left(z\right)$: Airy modulus function, $k$: 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		$$\varphi \left(-2^{\frac{1}{3}}{\alpha }^{\prime }\right)=\pi t,$$
ⓘ Defines: ${\alpha }^{\prime }$: coefficients (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\varphi \left(z\right)$: 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		${\alpha }_0^{\prime }$	$=1,$
		${\alpha }_1^{\prime }$	$={\alpha }^{\prime },$
		${\alpha }_2^{\prime }$	$=\frac{3}{10}\alpha _{}^{\prime }{}_{}{}^2-\frac{1}{10}\alpha _{}^{\prime }{}_{}{}^{-1},$
		${\alpha }_3^{\prime }$	$=-\frac{1}{350}\alpha _{}^{\prime }{}_{}{}^3-\frac{1}{25}-\frac{1}{200}\alpha _{}^{\prime }{}_{}{}^{-3},$
		${\alpha }_4^{\prime }$	$=-\frac{479}{63000}\alpha _{}^{\prime }{}_{}{}^4+\frac{509}{31500}{\alpha }^{\prime }+\frac{1}{1500}\alpha _{}^{\prime }{}_{}{}^{-2}-\frac{1}{2000}\alpha _{}^{\prime }{}_{}{}^{-5},$
ⓘ 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		${\beta }_0^{\prime }$	$=1,$
		${\beta }_1^{\prime }$	$=-\frac{1}{5}{\alpha }^{\prime },$
		${\beta }_2^{\prime }$	$=\frac{9}{350}\alpha _{}^{\prime }{}_{}{}^2+\frac{1}{100}\alpha _{}^{\prime }{}_{}{}^{-1},$
		${\beta }_3^{\prime }$	$=\frac{89}{15750}\alpha _{}^{\prime }{}_{}{}^3-\frac{47}{4500}+\frac{1}{3000}\alpha _{}^{\prime }{}_{}{}^{-3}.$
ⓘ 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		$$j_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$y_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$J_{\nu }^{\prime }\left(j_{\nu ,m}\right)\sim {(-1)}^m\frac{{(2/\nu )}^{\frac{2}{3}}}{\pi M\left(a_m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $a_k$: $k$th zero of Airy $\mathrm{Ai}$, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$Y_{\nu }^{\prime }\left(y_{\nu ,m}\right)\sim {(-1)}^{m-1}\frac{{(2/\nu )}^{\frac{2}{3}}}{\pi M\left(b_m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k}{{\nu }^{2k/3}},$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $b_k$: $k$th zero of Airy $\mathrm{Bi}$, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$j_{\nu ,m}^{\prime }\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$y_{\nu ,m}^{\prime }\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $m$: integer, $k$: 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		$$J_{\nu }\left(j_{\nu ,m}^{\prime }\right)\sim {(-1)}^{m-1}\frac{{(2/\nu )}^{\frac{1}{3}}}{\pi N\left(a_m^{\prime }\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k^{\prime }}{{\nu }^{2k/3}},$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N\left(z\right)$: Airy modulus function, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$, $m$: integer, $k$: 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		$$Y_{\nu }\left(y_{\nu ,m}^{\prime }\right)\sim {(-1)}^{m-1}\frac{{(2/\nu )}^{\frac{1}{3}}}{\pi N\left(b_m^{\prime }\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k^{\prime }}{{\nu }^{2k/3}}.$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N\left(z\right)$: Airy modulus function, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$, $b_k^{\prime }$: $k$th zero of Airy ${\mathrm{Bi}}^{\prime }$, $m$: integer, $k$: 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 $a_m$, $b_m$, $a_m^{\prime }$, $b_m^{\prime }$ are the $m$th negative zeros of $\mathrm{Ai}\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, respectively (§9.9), ${\alpha }_k$, ${\beta }_k$, ${\alpha }_k^{\prime }$, ${\beta }_k^{\prime }$ 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 $J_{\nu }^{\prime }\left(j_{\nu ,m}\right)$, $\alpha =-2^{-\frac{1}{3}}{b}_m$ in the case of $y_{\nu ,m}$ and $Y_{\nu }^{\prime }\left(y_{\nu ,m}\right)$, ${\alpha }^{\prime }=-2^{-\frac{1}{3}}{a}_m^{\prime }$ in the case of $j_{\nu ,m}^{\prime }$ and $J_{\nu }\left(j_{\nu ,m}^{\prime }\right)$, ${\alpha }^{\prime }=-2^{-\frac{1}{3}}{b}_m^{\prime }$ in the case of $y_{\nu ,m}^{\prime }$ and $Y_{\nu }\left(y_{\nu ,m}^{\prime }\right)$.