§10.49 Explicit Formulas

Define $a_k(\nu )$ as in (10.17.1). Then

10.49.1
ⓘ Symbols: $!$: factorial (as in $n!$), $n$: integer, $k$: nonnegative integer and $a_k(\nu )$: polynomial coefficient Referenced by: §10.49(ii), §10.50 Permalink: http://dlmf.nist.gov/10.49.E1 Encodings: TeX, pMML, png See also: Annotations for §10.49(i), §10.49 and Ch.10

10.49.2		$${\mathsf{j}}_n\left(z\right)=\begin{array}{l}\sin \left(z-\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+1}}\\ +\cos \left(z-\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+2}}.\end{array}$$
ⓘ 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, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.1.8 Referenced by: §10.50, §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E2 Encodings: TeX, pMML, png See also: Annotations for §10.49(i), §10.49 and Ch.10

10.49.3		${\mathsf{j}}_0\left(z\right)$	$={\displaystyle \frac{\sin z}{z}},$
		${\mathsf{j}}_1\left(z\right)$	$={\displaystyle \frac{\sin z}{z^2}}-{\displaystyle \frac{\cos z}{z}},$
		${\mathsf{j}}_2\left(z\right)$	$=\left(-{\displaystyle \frac{1}{z}}+{\displaystyle \frac{3}{z^3}}\right)\sin z-{\displaystyle \frac{3}{z^2}}\cos z.$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind and $z$: complex variable A&S Ref: 10.1.11 (corrected) Referenced by: §10.49(iii), §10.56 Permalink: http://dlmf.nist.gov/10.49.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(i), §10.49 and Ch.10

10.49.4		$${\mathsf{y}}_n\left(z\right)=\begin{array}{l}-\cos \left(z-\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+1}}\\ +\sin \left(z-\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+2}}.\end{array}$$
ⓘ 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, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.1.9 Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E4 Encodings: TeX, pMML, png See also: Annotations for §10.49(i), §10.49 and Ch.10

10.49.5		${\mathsf{y}}_0\left(z\right)$	$=-{\displaystyle \frac{\cos z}{z}},$
		${\mathsf{y}}_1\left(z\right)$	$=-{\displaystyle \frac{\cos z}{z^2}}-{\displaystyle \frac{\sin z}{z}},$
		${\mathsf{y}}_2\left(z\right)$	$=\left({\displaystyle \frac{1}{z}}-{\displaystyle \frac{3}{z^3}}\right)\cos z-{\displaystyle \frac{3}{z^2}}\sin z.$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind and $z$: complex variable A&S Ref: 10.1.12 (corrected) Referenced by: §10.50, §10.56 Permalink: http://dlmf.nist.gov/10.49.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(i), §10.49 and Ch.10

10.49.6		${\mathsf{h}}_n^{(1)}\left(z\right)$	$={\mathrm{e}}^{\mathrm{i}z}{\displaystyle \sum _{k=0}^n}{\mathrm{i}}^{k-n-1}{\displaystyle \frac{a_k(n+\frac{1}{2})}{z^{k+1}}},$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\mathsf{h}}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.1.16 Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E6 Encodings: TeX, pMML, png See also: Annotations for §10.49(i), §10.49 and Ch.10
10.49.7		${\mathsf{h}}_n^{(2)}\left(z\right)$	$={\mathrm{e}}^{-\mathrm{i}z}{\displaystyle \sum _{k=0}^n}{(-\mathrm{i})}^{k-n-1}{\displaystyle \frac{a_k(n+\frac{1}{2})}{z^{k+1}}}.$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\mathsf{h}}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.1.17 (corrected) Permalink: http://dlmf.nist.gov/10.49.E7 Encodings: TeX, pMML, png See also: Annotations for §10.49(i), §10.49 and Ch.10

Again, with $a_k(n+\frac{1}{2})$ as in (10.49.1),

10.49.8		$${\mathsf{i}}_n^{(1)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^{n+1}\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.2.9 (modified) Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E8 Encodings: TeX, pMML, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.9		${\mathsf{i}}_0^{(1)}\left(z\right)$	$={\displaystyle \frac{\sinh z}{z}},$
		${\mathsf{i}}_1^{(1)}\left(z\right)$	$=-{\displaystyle \frac{\sinh z}{z^2}}+{\displaystyle \frac{\cosh z}{z}},$
		${\mathsf{i}}_2^{(1)}\left(z\right)$	$=\left({\displaystyle \frac{1}{z}}+{\displaystyle \frac{3}{z^3}}\right)\sinh z-{\displaystyle \frac{3}{z^2}}\cosh z.$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.13 Permalink: http://dlmf.nist.gov/10.49.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.10		$${\mathsf{i}}_n^{(2)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^n\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.2.10 (modified) Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E10 Encodings: TeX, pMML, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.11		${\mathsf{i}}_0^{(2)}\left(z\right)$	$={\displaystyle \frac{\cosh z}{z}},$
		${\mathsf{i}}_1^{(2)}\left(z\right)$	$=-{\displaystyle \frac{\cosh z}{z^2}}+{\displaystyle \frac{\sinh z}{z}},$
		${\mathsf{i}}_2^{(2)}\left(z\right)$	$=\left({\displaystyle \frac{1}{z}}+{\displaystyle \frac{3}{z^3}}\right)\cosh z-{\displaystyle \frac{3}{z^2}}\sinh z.$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.14 Permalink: http://dlmf.nist.gov/10.49.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.12		$${\mathsf{k}}_n\left(z\right)=\frac{1}{2}\pi {\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient A&S Ref: 10.2.15 Referenced by: §10.52(ii) Permalink: http://dlmf.nist.gov/10.49.E12 Encodings: TeX, pMML, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.13		${\mathsf{k}}_0\left(z\right)$	$=\frac{1}{2}\pi {\displaystyle \frac{{\mathrm{e}}^{-z}}{z}},$
		${\mathsf{k}}_1\left(z\right)$	$=\frac{1}{2}\pi {\mathrm{e}}^{-z}\left({\displaystyle \frac{1}{z}}+{\displaystyle \frac{1}{z^2}}\right),$
		${\mathsf{k}}_2\left(z\right)$	$=\frac{1}{2}\pi {\mathrm{e}}^{-z}\left({\displaystyle \frac{1}{z}}+{\displaystyle \frac{3}{z^2}}+{\displaystyle \frac{3}{z^3}}\right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.17 Permalink: http://dlmf.nist.gov/10.49.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(ii), §10.49 and Ch.10

${\sum }_{k=0}^n{a}_k(n+\frac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

10.49.14		${\mathsf{j}}_n\left(z\right)$	$=z^n{\left(-{\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^n{\displaystyle \frac{\sin z}{z}},$
		${\mathsf{y}}_n\left(z\right)$	$=-z^n{\left(-{\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^n{\displaystyle \frac{\cos z}{z}}.$
ⓘ Symbols: $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\sin z$: 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 and $z$: complex variable A&S Ref: 10.1.25, 10.1.26 Referenced by: §10.49(iii) Permalink: http://dlmf.nist.gov/10.49.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.49(iii), §10.49 and Ch.10

10.49.15		${\mathsf{i}}_n^{(1)}\left(z\right)$	$=z^n{\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^n{\displaystyle \frac{\sinh z}{z}},$
		${\mathsf{i}}_n^{(2)}\left(z\right)$	$=z^n{\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^n{\displaystyle \frac{\cosh z}{z}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable A&S Ref: 10.2.24, 10.2 25 Permalink: http://dlmf.nist.gov/10.49.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.49(iii), §10.49 and Ch.10

10.49.16		$${\mathsf{k}}_n\left(z\right)={(-1)}^n\frac{1}{2}\pi z^n{\left(\frac{1}{z}\frac{d}{dz}\right)}^n\frac{{\mathrm{e}}^{-z}}{z}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/10.49.E16 Encodings: TeX, pMML, png See also: Annotations for §10.49(iii), §10.49 and Ch.10

Denote

10.49.17		$$s_k(n+\frac{1}{2})=\frac{(2k)!(n+k)!}{2^{2k}{(k!)}^2(n-k)!},$$
$k=0,1,\mathrm{\dots },n$.
ⓘ Defines: $s_k(n)$ (locally) Symbols: $!$: factorial (as in $n!$), $n$: integer and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/10.49.E17 Encodings: TeX, pMML, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

Then

10.49.18		$${\mathsf{j}}_n^2\left(z\right)+{\mathsf{y}}_n^2\left(z\right)=\sum _{k=0}^n\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$
ⓘ 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, $k$: nonnegative integer, $z$: complex variable and $s_k(n)$ A&S Ref: 10.1.27 Referenced by: §10.49(iv) Permalink: http://dlmf.nist.gov/10.49.E18 Encodings: TeX, pMML, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.19		${\mathsf{j}}_0^2\left(z\right)+{\mathsf{y}}_0^2\left(z\right)$	$=z^{-2},$
		${\mathsf{j}}_1^2\left(z\right)+{\mathsf{y}}_1^2\left(z\right)$	$=z^{-2}+z^{-4},$
		${\mathsf{j}}_2^2\left(z\right)+{\mathsf{y}}_2^2\left(z\right)$	$=z^{-2}+3z^{-4}+9z^{-6}.$
ⓘ 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 and $z$: complex variable A&S Ref: 10.1.28--10.1.30 Permalink: http://dlmf.nist.gov/10.49.E19 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.20		$${\left({\mathsf{i}}_n^{(1)}\left(z\right)\right)}^2-{\left({\mathsf{i}}_n^{(2)}\left(z\right)\right)}^2={(-1)}^{n+1}\sum _{k=0}^n{(-1)}^k\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$
ⓘ Symbols: ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $s_k(n)$ A&S Ref: 10.2.26 Referenced by: §10.49(iv) Permalink: http://dlmf.nist.gov/10.49.E20 Encodings: TeX, pMML, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.21		${\left({\mathsf{i}}_0^{(1)}\left(z\right)\right)}^2-{\left({\mathsf{i}}_0^{(2)}\left(z\right)\right)}^2$	$=-z^{-2},$
		${\left({\mathsf{i}}_1^{(1)}\left(z\right)\right)}^2-{\left({\mathsf{i}}_1^{(2)}\left(z\right)\right)}^2$	$=z^{-2}-z^{-4},$
		${\left({\mathsf{i}}_2^{(1)}\left(z\right)\right)}^2-{\left({\mathsf{i}}_2^{(2)}\left(z\right)\right)}^2$	$=-z^{-2}+3z^{-4}-9z^{-6}.$
ⓘ Symbols: ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function and $z$: complex variable A&S Ref: 10.2.27--10.2.29 Permalink: http://dlmf.nist.gov/10.49.E21 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(iv), §10.49 and Ch.10