§10.47 Definitions and Basic Properties

Referenced by:: §10.1, §10.47(i)
Permalink:: http://dlmf.nist.gov/10.47

§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

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

10.47.1 $z^{2}\frac{{d}^{2}w}{{dz}^{2}}+2z\frac{dw}{dz}+\left(z^{2}-n(n+1)\right)w=0,$

Symbols:: $\frac{df}{dx}$ : 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
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10.47.2 $z^{2}\frac{{d}^{2}w}{{dz}^{2}}+2z\frac{dw}{dz}-\left(z^{2}+n(n+1)\right)w=0.$

Symbols:: $\frac{df}{dx}$ : 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)
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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

Notes:: For (10.47.3)–(10.47.9) use (10.2.3), (10.4.6), (10.27.3).
Keywords:: differential equations, spherical Bessel functions
Referenced by:: ¶ ‣ §1.17(ii), ¶ ‣ §10.16, ¶ ‣ §10.39, §10.75(ix), §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

¶ Equation (10.47.1)

Keywords:: spherical Bessel functions

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:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind
Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $n$ : 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
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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:: $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind
Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.49(iv), §10.53, §10.60(ii)
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10.47.5 $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{{(1)}}_{{n+\frac{1}{2}}}}\/}\nolimits\!\left(z\right)=(-1)^{{n+1}}i\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{{(1)}}_{{-n-\frac{1}{2}}}}\/}\nolimits\!\left(z\right),$

Defines:: $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind
Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $n$ : integer and $z$ : complex variable
Referenced by:: §10.51(i)
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10.47.6 $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{{(2)}}_{{n+\frac{1}{2}}}}\/}\nolimits\!\left(z\right)=(-1)^{n}i\sqrt{\tfrac{1}{2}\pi/z}\mathop{{H^{{(2)}}_{{-n-\frac{1}{2}}}}\/}\nolimits\!\left(z\right).$

Defines:: $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind
Symbols:: $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $n$ : integer and $z$ : complex variable
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$\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ 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)

Keywords:: spherical Bessel functions

10.47.7 $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}\mathop{I_{{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right)$

Defines:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function
Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : 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)
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10.47.8 $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi/z}\mathop{I_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(z\right)$

Defines:: $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function
Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.2.2 (modified)
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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:: $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function
Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : 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
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$\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , 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

Notes:: Use §10.52.
Keywords:: differential equations, spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.47.iii

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

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

10.47.10

$\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)+i\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right),$

$\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)-i\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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
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10.47.11 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=(-1)^{{n+1}}\tfrac{1}{2}\pi\left(\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)-\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : 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
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10.47.12

$\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=i^{{-n}}\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(iz\right),$

$\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=i^{{-n-1}}\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(iz\right).$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : 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)
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10.47.13 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=-\tfrac{1}{2}\pi i^{n}\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(iz\right)=-\tfrac{1}{2}\pi i^{{-n}}\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(-iz\right).$

Symbols:: $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.49(ii), §10.60(i)
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§10.47(v) Reflection Formulas

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).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.47.v

10.47.14 $\displaystyle \mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(-z\right) =(-1)^{n}\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right),$ $\displaystyle \mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(-z\right) =(-1)^{{n+1}}\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.59
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10.47.15 $\displaystyle \mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(-z\right) =(-1)^{n}\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right),$ $\displaystyle \mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(-z\right) =(-1)^{n}\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable
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10.47.16 $\displaystyle \mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(-z\right) =(-1)^{n}\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right),$ $\displaystyle \mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(-z\right) =(-1)^{{n+1}}\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Referenced by:: §10.47(v)
Permalink:: http://dlmf.nist.gov/10.47.E16
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10.47.17 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(-z\right)=-\tfrac{1}{2}\pi\left(\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)+\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Referenced by:: §10.47(v)
Permalink:: http://dlmf.nist.gov/10.47.E17
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