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
Referenced by:: §10.75(x)
Permalink:: http://dlmf.nist.gov/10.58

For $n\geq 0$ the th positive zeros of $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ , and ${\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ 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 ${\mathop{\mathsf{j}_{{0}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ .

With the notation of §10.21(i),

10.58.1

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

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

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\mathop{j_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{y_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)$ , : integer, : integer, : real variable, $a_{{n,m}}$ : ^th zero of $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , $a^{{\prime}}_{{n,m}}$ : ^th zero of ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $b_{{n,m}}$ : ^th zero of $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ and $b^{{\prime}}_{{n,m}}$ : ^th zero of ${\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$
Permalink:: http://dlmf.nist.gov/10.58.E1
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10.58.2

${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(a_{{n,m}}\right)=% \sqrt{\frac{\pi}{2\mathop{j_{{n+\frac{1}{2},m}}\/}\nolimits}}{\mathop{J_{{n+% \frac{1}{2}}}\/}\nolimits^{{\prime}}}\!\left(\mathop{j_{{n+\frac{1}{2},m}}\/}% \nolimits\right),$

${\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{\prime}}}\!\left(b_{{n,m}}\right)=% \sqrt{\frac{\pi}{2\mathop{y_{{n+\frac{1}{2},m}}\/}\nolimits}}{\mathop{Y_{{n+% \frac{1}{2}}}\/}\nolimits^{{\prime}}}\!\left(\mathop{y_{{n+\frac{1}{2},m}}\/}% \nolimits\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\mathop{j_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{y_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)$ , : integer, : integer, $a_{{n,m}}$ : ^th zero of $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ and $b_{{n,m}}$ : ^th zero of $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$
Permalink:: http://dlmf.nist.gov/10.58.E2
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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).

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