§10.47 Definitions and Basic Properties

§10.47(i) Differential Equations

 10.47.1 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}w}{\mathrm{% d}z}+\left(z^{2}-n(n+1)\right)w=0,$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $n$: integer and $z$: complex variable A&S Ref: 10.1.1 Referenced by: §10.47(ii), §10.47(i), §10.47(iii), §30.2(iii) Permalink: http://dlmf.nist.gov/10.47.E1 Encodings: TeX, pMML, png See also: Annotations for §10.47(i), §10.47 and Ch.10
 10.47.2 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}w}{\mathrm{% d}z}-\left(z^{2}+n(n+1)\right)w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $n$: integer and $z$: complex variable A&S Ref: 10.2.1 Referenced by: §10.47(ii), §10.47(i), §10.47(iii) Permalink: http://dlmf.nist.gov/10.47.E2 Encodings: TeX, pMML, png See also: Annotations for §10.47(i), §10.47 and Ch.10

Here, and throughout the remainder of §§10.4710.60, $n$ is a nonnegative integer. (This is in contrast to other treatments of spherical Bessel functions, including Abramowitz and Stegun (1964, Chapter 10), in which $n$ can be any integer. However, there is a gain in symmetry, without any loss of generality in applications, on restricting $n\geq 0$.)

Equations (10.47.1) and (10.47.2) each have a regular singularity at $z=0$ with indices $n$, $-n-1$, and an irregular singularity at $z=\infty$ of rank $1$; compare §§2.7(i)2.7(ii).

§10.47(ii) Standard Solutions

Equation (10.47.1)

 10.47.3 $\mathsf{j}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}J_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}Y_{-n-\frac{1}{2}}\left(z\right),$ ⓘ Defines: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the first kind Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $z$: complex variable A&S Ref: 10.1.15 Referenced by: §1.17(iv), §10.47(ii), §10.47(iv), §10.47(v), §10.49(iv), §10.51(i), §10.53, §10.54, §10.57, §10.57, §10.59, §10.59, §10.60(ii), §11.4(i), §33.9(i) Permalink: http://dlmf.nist.gov/10.47.E3 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10
 10.47.4 $\mathsf{y}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}Y_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}J_{-n-\frac{1}{2}}\left(z\right),$ ⓘ Defines: $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$: spherical Bessel function of the second kind Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $z$: complex variable Referenced by: §10.49(iv), §10.53, §10.60(ii) Permalink: http://dlmf.nist.gov/10.47.E4 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10
 10.47.5 ${\mathsf{h}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{n+\frac% {1}{2}}}\left(z\right)=(-1)^{n+1}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{-% n-\frac{1}{2}}}\left(z\right),$ ⓘ Defines: ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$: spherical Bessel function of the third kind Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable Referenced by: §10.51(i) Permalink: http://dlmf.nist.gov/10.47.E5 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10
 10.47.6 ${\mathsf{h}^{(2)}_{n}}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{n+\frac% {1}{2}}}\left(z\right)=(-1)^{n}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{-n-% \frac{1}{2}}}\left(z\right).$ ⓘ Defines: ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$: spherical Bessel function of the third kind Symbols: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/10.47.E6 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10

$\mathsf{j}_{n}\left(z\right)$ and $\mathsf{y}_{n}\left(z\right)$ are the spherical Bessel functions of the first and second kinds, respectively; ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ and ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ are the spherical Bessel functions of the third kind.

Equation (10.47.2)

 10.47.7 $\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi/z}I_{n+\frac{1}{2}}\left(z\right)$ ⓘ Defines: ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.2.2 (modified) Referenced by: §10.51(ii), §10.53, §10.57, §11.4(i), §7.6(ii) Permalink: http://dlmf.nist.gov/10.47.E7 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10 10.47.8 $\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi/z}I_{-n-\frac{1}{2}}\left(z\right)$ ⓘ Defines: ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.2.2 (modified) Permalink: http://dlmf.nist.gov/10.47.E8 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10
 10.47.9 $\mathsf{k}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}K_{n+\frac{1}{2}}\left(z% \right)=\sqrt{\tfrac{1}{2}\pi/z}K_{-n-\frac{1}{2}}\left(z\right).$ ⓘ Defines: $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $n$: integer and $z$: complex variable Referenced by: §10.47(ii), §10.47(iv), §10.47(v), §10.51(ii), §10.54, §10.57, §10.59 Permalink: http://dlmf.nist.gov/10.47.E9 Encodings: TeX, pMML, png See also: Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10

${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, and $\mathsf{k}_{n}\left(z\right)$ are the modified spherical Bessel functions.

Many properties of $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$, ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, and $\mathsf{k}_{n}\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, $z^{-n}\mathsf{j}_{n}\left(z\right)$, $z^{n+1}\mathsf{y}_{n}\left(z\right)$, $z^{n+1}{\mathsf{h}^{(1)}_{n}}\left(z\right)$, $z^{n+1}{\mathsf{h}^{(2)}_{n}}\left(z\right)$, $z^{-n}{\mathsf{i}^{(1)}_{n}}\left(z\right)$, $z^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right)$, and $z^{n+1}\mathsf{k}_{n}\left(z\right)$ are all entire functions of $z$.

§10.47(iii) Numerically Satisfactory Pairs of Solutions

For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $\mathsf{j}$, $\mathsf{y}$, $\mathsf{h}$, and $n$, respectively.

For (10.47.2) numerically satisfactory pairs of solutions are ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(z\right)$ in the right half of the $z$-plane, and ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(-z\right)$ in the left half of the $z$-plane.

§10.47(iv) Interrelations

 10.47.10 $\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$ $\displaystyle=\mathsf{j}_{n}\left(z\right)+i\mathsf{y}_{n}\left(z\right),$ $\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$ $\displaystyle=\mathsf{j}_{n}\left(z\right)-i\mathsf{y}_{n}\left(z\right).$
 10.47.11 $\mathsf{k}_{n}\left(z\right)=(-1)^{n+1}\tfrac{1}{2}\pi\left({\mathsf{i}^{(1)}_% {n}}\left(z\right)-{\mathsf{i}^{(2)}_{n}}\left(z\right)\right).$
 10.47.12 $\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$ $\displaystyle=i^{-n}\mathsf{j}_{n}\left(iz\right),$ $\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$ $\displaystyle=i^{-n-1}\mathsf{y}_{n}\left(iz\right).$
 10.47.13 $\mathsf{k}_{n}\left(z\right)=-\tfrac{1}{2}\pi i^{n}{\mathsf{h}^{(1)}_{n}}\left% (iz\right)=-\tfrac{1}{2}\pi i^{-n}{\mathsf{h}^{(2)}_{n}}\left(-iz\right).$

§10.47(v) Reflection Formulas

 10.47.14 $\displaystyle\mathsf{j}_{n}\left(-z\right)$ $\displaystyle=(-1)^{n}\mathsf{j}_{n}\left(z\right),$ $\displaystyle\mathsf{y}_{n}\left(-z\right)$ $\displaystyle=(-1)^{n+1}\mathsf{y}_{n}\left(z\right),$ 10.47.15 $\displaystyle{\mathsf{h}^{(1)}_{n}}\left(-z\right)$ $\displaystyle=(-1)^{n}{\mathsf{h}^{(2)}_{n}}\left(z\right),$ $\displaystyle{\mathsf{h}^{(2)}_{n}}\left(-z\right)$ $\displaystyle=(-1)^{n}{\mathsf{h}^{(1)}_{n}}\left(z\right).$ 10.47.16 $\displaystyle{\mathsf{i}^{(1)}_{n}}\left(-z\right)$ $\displaystyle=(-1)^{n}{\mathsf{i}^{(1)}_{n}}\left(z\right),$ $\displaystyle{\mathsf{i}^{(2)}_{n}}\left(-z\right)$ $\displaystyle=(-1)^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right),$
 10.47.17 $\mathsf{k}_{n}\left(-z\right)=-\tfrac{1}{2}\pi\left({\mathsf{i}^{(1)}_{n}}% \left(z\right)+{\mathsf{i}^{(2)}_{n}}\left(z\right)\right).$