# §10.58 Zeros

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 $\displaystyle a_{n,m}$ $\displaystyle=j_{n+\frac{1}{2},m},$ $\displaystyle b_{n,m}$ $\displaystyle=y_{n+\frac{1}{2},m},$ ⓘ 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 10
 10.58.2 $\displaystyle\mathsf{j}_{n}'\left(a_{n,m}\right)$ $\displaystyle=\sqrt{\frac{\pi}{2j_{n+\frac{1}{2},m}}}J_{n+\frac{1}{2}}'\left(j% _{n+\frac{1}{2},m}\right),$ $\displaystyle\mathsf{y}_{n}'\left(b_{n,m}\right)$ $\displaystyle=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{n+\frac{1}{2}}'\left(y% _{n+\frac{1}{2},m}\right).$

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).