10 Bessel Functions Spherical Bessel Functions 10.49 Explicit Formulas 10.51 Recurrence Relations and Derivatives

§10.50 Wronskians and Cross-Products

Notes:: That the Wronskians are constant multiples of $z^{{-2}}$ follows from (1.13.5). The constants can be found from the limiting forms (and their derivatives) given in §§10.52(i) or 10.52(ii). For (10.50.3) combine (10.50.1) with (10.51.1) and (10.51.2). For (10.50.4) use (10.49.2)–(10.49.5).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.50

10.50.1

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{j}_{{n}}\/}\nolimits\!% \left(z\right),\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)\right\}=z^% {{-2}},$

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}% \nolimits\!\left(z\right),\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!% \left(z\right)\right\}=-2iz^{{-2}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, : integer and : complex variable
A&S Ref:: 10.1.6, 10.1.7
Referenced by:: §10.50
Permalink:: http://dlmf.nist.gov/10.50.E1
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10.50.2

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}% \nolimits\!\left(z\right),\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!% \left(z\right)\right\}=(-1)^{{n+1}}z^{{-2}},$

$\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}% \nolimits\!\left(z\right),\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)% \right\}=\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{{(2)}}_{{n% }}}\/}\nolimits\!\left(z\right),\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z% \right)\right\}\\ =-\tfrac{1}{2}\pi z^{{-2}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer and : complex variable
A&S Ref:: 10.2.7, 10.2.8
Permalink:: http://dlmf.nist.gov/10.50.E2
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10.50.3

$\mathop{\mathsf{j}_{{n+1}}\/}\nolimits\!\left(z\right)\mathop{\mathsf{y}_{{n}}% \/}\nolimits\!\left(z\right)-\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z% \right)\mathop{\mathsf{y}_{{n+1}}\/}\nolimits\!\left(z\right)=z^{{-2}},$

$\mathop{\mathsf{j}_{{n+2}}\/}\nolimits\!\left(z\right)\mathop{\mathsf{y}_{{n}}% \/}\nolimits\!\left(z\right)-\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z% \right)\mathop{\mathsf{y}_{{n+2}}\/}\nolimits\!\left(z\right)=(2n+3)z^{{-3}}.$

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, : integer and : complex variable
A&S Ref:: 10.2.31, 10.1.32
Referenced by:: §10.50, §10.50
Permalink:: http://dlmf.nist.gov/10.50.E3
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10.50.4 $\mathop{\mathsf{j}_{{0}}\/}\nolimits\!\left(z\right)\mathop{\mathsf{j}_{{n}}\/% }\nolimits\!\left(z\right)+\mathop{\mathsf{y}_{{0}}\/}\nolimits\!\left(z\right% )\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits% \!\left(\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor n/2\right\rfloor}}(% -1)^{k}\frac{a_{{2k}}(n+\tfrac{1}{2})}{z^{{2k+2}}}+\mathop{\sin\/}\nolimits\!% \left(\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor(n-1)/2\right\rfloor}}% (-1)^{k}\frac{a_{{2k+1}}(n+\tfrac{1}{2})}{z^{{2k+3}}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.1.33 (modified)
Referenced by:: §10.50, §10.50
Permalink:: http://dlmf.nist.gov/10.50.E4
Encodings:: TeX, pMML, png

where $a_{k}(n+\tfrac{1}{2})$ is given by (10.49.1).

Results corresponding to (10.50.3) and (10.50.4) for $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ and $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ are obtainable via (10.47.12).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.49 Explicit Formulas 10.51 Recurrence Relations and Derivatives