spherical Bessel functions

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1: 10.48 Graphs

§10.48 Graphs

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Figure 10.48.7: ${\mathsf{i}}_5^{(1)}\left(x\right)$, ${\mathsf{i}}_5^{(2)}\left(x\right)$, ${\mathsf{k}}_5\left(x\right)$, $0\le x\le 8$. Magnify

2: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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10.50.4 $${\mathsf{j}}_0\left(z\right){\mathsf{j}}_n\left(z\right)+{\mathsf{y}}_0\left(z\right){\mathsf{y}}_n\left(z\right)=\cos \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+2}}+\sin \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+3}},$$

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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, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.33 (modified)

Referenced by:

§10.50, §10.50

Permalink:

http://dlmf.nist.gov/10.50.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.50 and Ch.10

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3: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, or ${\mathsf{h}}_n^{(2)}\left(z\right)$. … ►

$$n{f}_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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$$f_n^{\prime }(z)=-f_{n+1}(z)+(n/z)f_n(z),$$ $n=0,1,\mathrm{\dots }.$

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$$n{g}_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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4: 10.47 Definitions and Basic Properties

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Equation (10.47.1)

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Equation (10.47.2)

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§10.47(iv) Interrelations

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5: 10.52 Limiting Forms

§10.52 Limiting Forms

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10.52.1 $${\mathsf{j}}_n\left(z\right),{\mathsf{i}}_n^{(1)}\left(z\right)\sim z^n/(2n+1)!!,$$

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Symbols:

$\sim$: asymptotic equality, $!!$: double factorial (as in $n!!$), ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.4

Permalink:

http://dlmf.nist.gov/10.52.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

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10.52.2 $$-{\mathsf{y}}_n\left(z\right),\mathrm{i}{\mathsf{h}}_n^{(1)}\left(z\right),-\mathrm{i}{\mathsf{h}}_n^{(2)}\left(z\right),{(-1)}^n{\mathsf{i}}_n^{(2)}\left(z\right),(2/\pi ){\mathsf{k}}_n\left(z\right)\sim (2n-1)!!/z^{n+1}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial (as in $n!!$), $\mathrm{i}$: imaginary unit, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, ${\mathsf{h}}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${\mathsf{h}}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.5

Permalink:

http://dlmf.nist.gov/10.52.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

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10.52.5 $${\mathsf{i}}_n^{(1)}\left(z\right)\sim {\mathsf{i}}_n^{(2)}\left(z\right)\sim \frac{1}{2}{z}^{-1}{\mathrm{e}}^z,$$ ,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.52.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(ii), §10.52 and Ch.10

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10.52.6 $${\mathsf{k}}_n\left(z\right)\sim \frac{1}{2}\pi z^{-1}{\mathrm{e}}^{-z}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: 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 and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.52.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(ii), §10.52 and Ch.10

6: 10.49 Explicit Formulas

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§10.49(i) Unmodified Functions

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§10.49(ii) Modified Functions

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§10.49(iii) Rayleigh’s Formulas

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§10.49(iv) Sums or Differences of Squares

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10.49.20 $${\left({\mathsf{i}}_n^{(1)}\left(z\right)\right)}^2-{\left({\mathsf{i}}_n^{(2)}\left(z\right)\right)}^2={(-1)}^{n+1}\sum _{k=0}^n{(-1)}^k\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$

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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:

pMML, png, TeX

See also:

Annotations for §10.49(iv), §10.49 and Ch.10

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7: 10.55 Continued Fractions

§10.55 Continued Fractions

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8: 10.60 Sums

§10.60 Sums

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§10.60(i) Addition Theorems

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§10.60(ii) Duplication Formulas

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►

§10.60(iv) Compendia

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9: 30.10 Series and Integrals

… ►For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

10: 10.76 Approximations

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Spherical Bessel Functions

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