10 Bessel Functions Spherical Bessel Functions 10.57 Uniform Asymptotic Expansions for Large Order 10.59 Integrals

§10.58 Zeros

ⓘ

Keywords:: derivatives, spherical Bessel functions, zeros
Referenced by:: §10.75(x)
Permalink:: http://dlmf.nist.gov/10.58
See also:: Annotations for Ch.10

For $n\geq 0$ the $m$ th positive zeros of $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , and $\mathsf{y}_{n}'\left(x\right)$ are denoted by $a_{n,m}$ , $a^{\prime}_{n,m}$ , $b_{n,m}$ , and $b^{\prime}_{n,m}$ , respectively, except that for $n=0$ we count $x=0$ as the first zero of $\mathsf{j}_{0}'\left(x\right)$ .

With the notation of §10.21(i),

10.58.1	$\displaystyle a_{n,m}$	$\displaystyle=j_{n+\frac{1}{2},m},$
10.58.1	$\displaystyle b_{n,m}$	$\displaystyle=y_{n+\frac{1}{2},m},$
ⓘ Defines: $a_{n,m}$ : $m$ th zero of $\mathsf{j}_{n}\left(x\right)$ (locally), $a^{\prime}_{n,m}$ : $m$ th zero of $\mathsf{j}_{n}'\left(x\right)$ (locally), $b_{n,m}$ : $m$ th zero of $\mathsf{y}_{n}\left(x\right)$ (locally) and $b^{\prime}_{n,m}$ : $m$ th zero of $\mathsf{y}_{n}'\left(x\right)$ (locally) Symbols: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , $m$ : integer, $n$ : integer and $x$ : real variable Permalink: http://dlmf.nist.gov/10.58.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.58 and Ch.10

10.58.2	$\displaystyle\mathsf{j}_{n}'\left(a_{n,m}\right)$	$\displaystyle=\sqrt{\frac{\pi}{2j_{n+\frac{1}{2},m}}}J_{n+\frac{1}{2}}'\left(j% _{n+\frac{1}{2},m}\right),$
10.58.2	$\displaystyle\mathsf{y}_{n}'\left(b_{n,m}\right)$	$\displaystyle=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{n+\frac{1}{2}}'\left(y% _{n+\frac{1}{2},m}\right).$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , $m$ : integer, $n$ : integer, $a_{n,m}$ : $m$ th zero of $\mathsf{j}_{n}\left(x\right)$ and $b_{n,m}$ : $m$ th zero of $\mathsf{y}_{n}\left(x\right)$ Permalink: http://dlmf.nist.gov/10.58.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.58 and Ch.10

Hence properties of $a_{n,m}$ and $b_{n,m}$ are derivable straightforwardly from results given in §§10.21(i)–10.21(iii), 10.21(vi)–10.21(viii), and 10.21(x). However, there are no simple relations that connect the zeros of the derivatives. For some properties of $a^{\prime}_{n,m}$ and $b^{\prime}_{n,m}$ , including asymptotic expansions, see Olver (1960, pp. xix–xxi).

See also Davies (1973), de Bruin et al. (1981a, b), and Gottlieb (1985).

10.57 Uniform Asymptotic Expansions for Large Order 10.59 Integrals

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§10.58 Zeros

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