# §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 of $f$ with respect to $x$, $n$: 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)
 10.47.2 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}w}{\mathrm{% d}z}-\left(z^{2}+n(n+1)\right)w=0.$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $n$: 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)

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 $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}% \mathop{J_{n+\frac{1}{2}}\/}\nolimits\!\left(z\right)=(-1)^{n}\sqrt{\tfrac{1}{% 2}\pi/z}\mathop{Y_{-n-\frac{1}{2}}\/}\nolimits\!\left(z\right),$ Defines: $\mathop{\mathsf{j}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the first kind Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.47.4 $\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}% \mathop{Y_{n+\frac{1}{2}}\/}\nolimits\!\left(z\right)=(-1)^{n+1}\sqrt{\tfrac{1% }{2}\pi/z}\mathop{J_{-n-\frac{1}{2}}\/}\nolimits\!\left(z\right),$ Defines: $\mathop{\mathsf{y}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the second kind Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.47.5 $\mathop{{\mathsf{h}^{(1)}_{n}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}% \pi/z}\mathop{{H^{(1)}_{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=(-1)^{n+1}% \mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{(1)}_{-n-\frac{1}{2}}}\/}% \nolimits\!\left(z\right),$ Defines: $\mathop{{\mathsf{h}^{(1)}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the third kind Symbols: $\mathop{{H^{(1)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $n$: 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)
 10.47.6 $\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}% \pi/z}\mathop{{H^{(2)}_{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=(-1)^{n}% \mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{(2)}_{-n-\frac{1}{2}}}\/}% \nolimits\!\left(z\right).$ Defines: $\mathop{{\mathsf{h}^{(2)}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the third kind Symbols: $\mathop{{H^{(2)}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/10.47.E6 Encodings: TeX, pMML, png See also: Annotations for 10.47(ii)

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

### Equation (10.47.2)

 10.47.7 $\displaystyle\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi/z}\mathop{I_{n+\frac{1}{2}}\/}\nolimits\!% \left(z\right)$ Defines: $\mathop{{\mathsf{i}^{(1)}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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) 10.47.8 $\displaystyle\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi/z}\mathop{I_{-n-\frac{1}{2}}\/}\nolimits\!% \left(z\right)$ Defines: $\mathop{{\mathsf{i}^{(2)}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.47.9 $\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}% \mathop{K_{n+\frac{1}{2}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}% \mathop{K_{-n-\frac{1}{2}}\/}\nolimits\!\left(z\right).$ Defines: $\mathop{\mathsf{k}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: modified spherical Bessel function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

Many properties of $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(z\right)$, $\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)$, $\mathop{{\mathsf{h}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$, $\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$, $\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$, $\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$, and $\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, $z^{-n}\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(z\right)$, $z^{n+1}\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)$, $z^{n+1}\mathop{{\mathsf{h}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$, $z^{n+1}\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$, $z^{-n}\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$, $z^{n+1}\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$, and $z^{n+1}\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z\right)$ 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 $\mathop{J\/}\nolimits$, $\mathop{Y\/}\nolimits$, $H$, and $\nu$ replaced by $\mathop{\mathsf{j}\/}\nolimits$, $\mathop{\mathsf{y}\/}\nolimits$, $\mathsf{h}$, and $n$, respectively.

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

## §10.47(iv) Interrelations

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

## §10.47(v) Reflection Formulas

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