10 Bessel Functions Spherical Bessel Functions 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function 10.48 Graphs

§10.47 Definitions and Basic Properties

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Referenced by:: §10.1, §10.47(i)
Permalink:: http://dlmf.nist.gov/10.47
See also:: Annotations for Ch.10

§10.47(i) Differential Equations
§10.47(ii) Standard Solutions
§10.47(iii) Numerically Satisfactory Pairs of Solutions
§10.47(iv) Interrelations
§10.47(v) Reflection Formulas

§10.47(i) Differential Equations

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Keywords:: differential equations, singularities, spherical Bessel functions
Referenced by:: §29.18(i), Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/10.47.i
See also:: Annotations for §10.47 and Ch.10

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 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 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), §10.47 and Ch.10

Here, and throughout the remainder of §§10.47–10.60, $n$ 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

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Keywords:: differential equations, spherical Bessel functions, standard solutions
Notes:: For (10.47.3)–(10.47.9) use (10.2.3), (10.4.6), (10.27.3).
Referenced by:: §1.17(ii), §10.16, §10.39, 1st item, §33.5(ii), §33.9(i), §6.10(ii), §7.6(ii), §8.21(vi), §8.7, Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/10.47.ii
See also:: Annotations for §10.47 and Ch.10

Equation (10.47.1)

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Keywords:: definitions, of the first, second, and third kinds, spherical Bessel functions
See also:: Annotations for §10.47(ii), §10.47 and Ch.10

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: §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)

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Keywords:: analytic properties, definitions, modified, spherical Bessel functions
See also:: Annotations for §10.47(ii), §10.47 and Ch.10

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, (18.34.2) 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

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Keywords:: differential equations, numerically satisfactory solutions, spherical Bessel functions
Notes:: Use §10.52.
Permalink:: http://dlmf.nist.gov/10.47.iii
See also:: Annotations for §10.47 and Ch.10

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

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Keywords:: interrelations, spherical Bessel functions
Notes:: Use (10.4.3), (10.27.4), (10.27.6), (10.27.8), and the definitions (10.47.3)–(10.47.9).
Referenced by:: §10.60(ii)
Permalink:: http://dlmf.nist.gov/10.47.iv
See also:: Annotations for §10.47 and Ch.10

10.47.10	$\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$	$\displaystyle=\mathsf{j}_{n}\left(z\right)+i\mathsf{y}_{n}\left(z\right),$
10.47.10	$\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$	$\displaystyle=\mathsf{j}_{n}\left(z\right)-i\mathsf{y}_{n}\left(z\right).$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable Referenced by: §10.53 Permalink: http://dlmf.nist.gov/10.47.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.47(iv), §10.47 and Ch.10

10.47.11		$\mathsf{k}_{n}\left(z\right)=(-1)^{n+1}\tfrac{1}{2}\pi\left({\mathsf{i}^{(1)}_% {n}}\left(z\right)-{\mathsf{i}^{(2)}_{n}}\left(z\right)\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable A&S Ref: 10.2.4 (modified) Referenced by: §10.47(v), §10.53, §10.56, §10.57 Permalink: http://dlmf.nist.gov/10.47.E11 Encodings: TeX, pMML, png See also: Annotations for §10.47(iv), §10.47 and Ch.10

10.47.12	$\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$	$\displaystyle=i^{-n}\mathsf{j}_{n}\left(iz\right),$
10.47.12	$\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$	$\displaystyle=i^{-n-1}\mathsf{y}_{n}\left(iz\right).$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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 Referenced by: §10.49(ii), §10.49(iv), §10.50, §10.56, §10.60(i) Permalink: http://dlmf.nist.gov/10.47.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.47(iv), §10.47 and Ch.10

10.47.13		$\mathsf{k}_{n}\left(z\right)=-\tfrac{1}{2}\pi i^{n}{\mathsf{h}^{(1)}_{n}}\left% (iz\right)=-\tfrac{1}{2}\pi i^{-n}{\mathsf{h}^{(2)}_{n}}\left(-iz\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable Referenced by: §10.49(ii), §10.60(i) Permalink: http://dlmf.nist.gov/10.47.E13 Encodings: TeX, pMML, png See also: Annotations for §10.47(iv), §10.47 and Ch.10

§10.47(v) Reflection Formulas

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Keywords:: reflection formulas, spherical Bessel functions
Notes:: Use (10.11.1), (10.11.2), (10.34.1), with $m=1$ in each case, and the definitions (10.47.3)–(10.47.9). For (10.47.17) use (10.47.11) and (10.47.16).
Permalink:: http://dlmf.nist.gov/10.47.v
See also:: Annotations for §10.47 and Ch.10

10.47.14	$\displaystyle\mathsf{j}_{n}\left(-z\right)$	$\displaystyle=(-1)^{n}\mathsf{j}_{n}\left(z\right),$	$\displaystyle\mathsf{y}_{n}\left(-z\right)$	$\displaystyle=(-1)^{n+1}\mathsf{y}_{n}\left(z\right),$
ⓘ 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 Referenced by: §10.59 Permalink: http://dlmf.nist.gov/10.47.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.47(v), §10.47 and Ch.10
10.47.15	$\displaystyle{\mathsf{h}^{(1)}_{n}}\left(-z\right)$	$\displaystyle=(-1)^{n}{\mathsf{h}^{(2)}_{n}}\left(z\right),$	$\displaystyle{\mathsf{h}^{(2)}_{n}}\left(-z\right)$	$\displaystyle=(-1)^{n}{\mathsf{h}^{(1)}_{n}}\left(z\right).$
ⓘ Symbols: ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.47.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.47(v), §10.47 and Ch.10
10.47.16	$\displaystyle{\mathsf{i}^{(1)}_{n}}\left(-z\right)$	$\displaystyle=(-1)^{n}{\mathsf{i}^{(1)}_{n}}\left(z\right),$	$\displaystyle{\mathsf{i}^{(2)}_{n}}\left(-z\right)$	$\displaystyle=(-1)^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right),$
ⓘ Symbols: ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Referenced by: §10.47(v) Permalink: http://dlmf.nist.gov/10.47.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.47(v), §10.47 and Ch.10

10.47.17		$\mathsf{k}_{n}\left(-z\right)=-\tfrac{1}{2}\pi\left({\mathsf{i}^{(1)}_{n}}% \left(z\right)+{\mathsf{i}^{(2)}_{n}}\left(z\right)\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Referenced by: §10.47(v) Permalink: http://dlmf.nist.gov/10.47.E17 Encodings: TeX, pMML, png See also: Annotations for §10.47(v), §10.47 and Ch.10

10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function 10.48 Graphs

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