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

§10.58 Zeros

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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 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 we count as the first zero of $\mathsf{j}_{0}'\left(x\right)$ .

With the notation of §10.21(i),

10.58.1

$a_{n,m}=j_{n+\frac{1}{2},m},$

$b_{n,m}=y_{n+\frac{1}{2},m},$

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Defines:: $a_{n,m}$ : th zero of $\mathsf{j}_{n}\left(x\right)$ (locally), $a^{\prime}_{n,m}$ : th zero of $\mathsf{j}_{n}'\left(x\right)$ (locally), $b_{n,m}$ : th zero of $\mathsf{y}_{n}\left(x\right)$ (locally) and $b^{\prime}_{n,m}$ : 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 $Y_{\nu}\left(x\right)$ , : integer, : integer and : 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

$\mathsf{j}_{n}'\left(a_{n,m}\right)=\sqrt{\frac{\pi}{2j_{n+\frac{1}{2},m}}}J_{% n+\frac{1}{2}}'\left(j_{n+\frac{1}{2},m}\right),$

$\mathsf{y}_{n}'\left(b_{n,m}\right)=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{% n+\frac{1}{2}}'\left(y_{n+\frac{1}{2},m}\right).$

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).

10.57 Uniform Asymptotic Expansions for Large Order 10.59 Integrals

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