§10.50 Wronskians and Cross-Products

10.50.1		$𝒲\left\{{\mathsf{j}}_n\left(z\right),{\mathsf{y}}_n\left(z\right)\right\}$	$=z^{-2},$
		$𝒲\left\{{\mathsf{h}}_n^{(1)}\left(z\right),{\mathsf{h}}_n^{(2)}\left(z\right)\right\}$	$=-2\mathrm{i}{z}^{-2}.$
ⓘ Symbols: $𝒲$: Wronskian, $\mathrm{i}$: imaginary unit, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, ${\mathsf{h}}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${\mathsf{h}}_n^{(2)}\left(z\right)$: 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		$𝒲\left\{{\mathsf{i}}_n^{(1)}\left(z\right),{\mathsf{i}}_n^{(2)}\left(z\right)\right\}$	$={(-1)}^{n+1}{z}^{-2},$
		$𝒲\left\{{\mathsf{i}}_n^{(1)}\left(z\right),{\mathsf{k}}_n\left(z\right)\right\}$	$=𝒲\left\{{\mathsf{i}}_n^{(2)}\left(z\right),{\mathsf{k}}_n\left(z\right)\right\}=-\frac{1}{2}\pi z^{-2}.$
ⓘ Symbols: $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{k}}_n\left(z\right)$: 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		${\mathsf{j}}_{n+1}\left(z\right){\mathsf{y}}_n\left(z\right)-{\mathsf{j}}_n\left(z\right){\mathsf{y}}_{n+1}\left(z\right)$	$=z^{-2},$
		${\mathsf{j}}_{n+2}\left(z\right){\mathsf{y}}_n\left(z\right)-{\mathsf{j}}_n\left(z\right){\mathsf{y}}_{n+2}\left(z\right)$	$=(2n+3)z^{-3}.$
ⓘ Symbols: ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: 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		$$\begin{array}{l}{\mathsf{j}}_0\left(z\right){\mathsf{j}}_n\left(z\right)+{\mathsf{y}}_0\left(z\right){\mathsf{y}}_n\left(z\right)\\ =\cos \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+2}}+\sin \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+3}},\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$, $\sin z$: sine function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: 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 $a_k(n+\frac{1}{2})$ is given by (10.49.1).

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