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

§10.50 Wronskians and Cross-Products

ⓘ

Keywords:: Wronskians, cross-products, spherical Bessel functions
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).
Permalink:: http://dlmf.nist.gov/10.50
See also:: Annotations for Ch.10

10.50.1	$\displaystyle\mathscr{W}\left\{\mathsf{j}_{n}\left(z\right),\mathsf{y}_{n}% \left(z\right)\right\}$	$\displaystyle=z^{-2},$
10.50.1	$\displaystyle\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z\right),{\mathsf{h% }^{(2)}_{n}}\left(z\right)\right\}$	$\displaystyle=-2iz^{-2}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable A&S Ref: 10.1.6, 10.1.7 Referenced by: §10.50 Permalink: http://dlmf.nist.gov/10.50.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.50 and Ch.10

10.50.2	$\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),{\mathsf{i% }^{(2)}_{n}}\left(z\right)\right\}$	$\displaystyle=(-1)^{n+1}z^{-2},$
10.50.2	$\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),\mathsf{k}% _{n}\left(z\right)\right\}$	$\displaystyle=\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z\right),\mathsf{k% }_{n}\left(z\right)\right\}=-\tfrac{1}{2}\pi z^{-2}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable A&S Ref: 10.2.7, 10.2.8 Permalink: http://dlmf.nist.gov/10.50.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.50 and Ch.10

10.50.3	$\displaystyle\mathsf{j}_{n+1}\left(z\right)\mathsf{y}_{n}\left(z\right)-% \mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+1}\left(z\right)$	$\displaystyle=z^{-2},$
10.50.3	$\displaystyle\mathsf{j}_{n+2}\left(z\right)\mathsf{y}_{n}\left(z\right)-% \mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+2}\left(z\right)$	$\displaystyle=(2n+3)z^{-3}.$
ⓘ 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, $n$ : integer and $z$ : 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.50 and Ch.10

10.50.4		$\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z\right)+\mathsf{y}_{0}\left(z% \right)\mathsf{y}_{n}\left(z\right)=\cos\left(\tfrac{1}{2}n\pi\right)\sum_{k=0% }^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2% }}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer, $z$ : complex variable and $a_{k}(\nu)$ : polynomial coefficient 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 See also: Annotations for §10.50 and Ch.10

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

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

10.49 Explicit Formulas 10.51 Recurrence Relations and Derivatives

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§10.50 Wronskians and Cross-Products

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