# §10.50 Wronskians and Cross-Products

 10.50.1 $\displaystyle\mathscr{W}\left\{\mathsf{j}_{n}\left(z\right),\mathsf{y}_{n}% \left(z\right)\right\}$ $\displaystyle=z^{-2},$ $\displaystyle\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z\right),{\mathsf{h% }^{(2)}_{n}}\left(z\right)\right\}$ $\displaystyle=-2iz^{-2}.$
 10.50.2 $\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),{\mathsf{i% }^{(2)}_{n}}\left(z\right)\right\}$ $\displaystyle=(-1)^{n+1}z^{-2},$ $\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),\mathsf{k}% _{n}\left(z\right)\right\}$ $\displaystyle=\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z\right),\mathsf{k% }_{n}\left(z\right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}.$
 10.50.3 $\displaystyle\mathsf{j}_{n+1}\left(z\right)\mathsf{y}_{n}\left(z\right)-% \mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+1}\left(z\right)$ $\displaystyle=z^{-2},$ $\displaystyle\mathsf{j}_{n+2}\left(z\right)\mathsf{y}_{n}\left(z\right)-% \mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+2}\left(z\right)$ $\displaystyle=(2n+3)z^{-3}.$ ⓘ 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 10
 10.50.4 $\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z\right)+\mathsf{y}_{0}\left(z% \right)\mathsf{y}_{n}\left(z\right)=\cos\left(\tfrac{1}{2}n\pi\right)\sum_{k=0% }^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2% }}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}},$

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).