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11: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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$$𝒲\left\{{𝗁}_n^{(1)}\left(z\right),{𝗁}_n^{(2)}\left(z\right)\right\}=-2\mathrm{i}{z}^{-2}.$$

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$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗂}_n^{(2)}\left(z\right)\right\}={(-1)}^{n+1}{z}^{-2},$$

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$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗄}_n\left(z\right)\right\}=𝒲\left\{{𝗂}_n^{(2)}\left(z\right),{𝗄}_n\left(z\right)\right\}=-\frac{1}{2}\pi z^{-2}.$$

… ►Results corresponding to (10.50.3) and (10.50.4) for ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗂}_n^{(2)}\left(z\right)$ are obtainable via (10.47.12).

12: 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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13: 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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${\left({𝗂}_0^{(1)}\left(z\right)\right)}^2-{\left({𝗂}_0^{(2)}\left(z\right)\right)}^2=-z^{-2},$

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14: 10.53 Power Series

§10.53 Power Series

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10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

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10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►For ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For ${𝗄}_n\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

15: 10.56 Generating Functions

§10.56 Generating Functions

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10.56.1 $$\frac{\cos \sqrt{z^2-2zt}}{z}=\frac{\cos z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗃}_{n-1}\left(z\right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E1

Encodings:

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See also:

Annotations for §10.56 and Ch.10

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10.56.2 $$\frac{\sin \sqrt{z^2-2zt}}{z}=\frac{\sin z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗒}_{n-1}\left(z\right).$$

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Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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16: 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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17: 10.51 Recurrence Relations and Derivatives

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

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

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

… ►Let $g_n(z)$ denote ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${𝗄}_n\left(z\right)$. Then …

18: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for ${𝗂}_n^{(2)}\left((n+\frac{1}{2})z\right)$ the connection formula (10.47.11) is available. ►For the corresponding expansion for ${𝗃}_n^{\prime }\left((n+\frac{1}{2})z\right)$ use ►

10.57.1 $${𝗃}_n^{\prime }\left((n+\frac{1}{2})z\right)=\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{1}{2}}}{J}_{n+\frac{1}{2}}^{\prime }\left((n+\frac{1}{2})z\right)-\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{3}{2}}}{J}_{n+\frac{1}{2}}\left((n+\frac{1}{2})z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.57

Permalink:

http://dlmf.nist.gov/10.57.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.57 and Ch.10

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19: 10.58 Zeros

§10.58 Zeros

►For $n\ge 0$ the $m$th positive zeros of ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, and ${𝗒}_n^{\prime }\left(x\right)$ are denoted by $a_{n,m}$, $a_{n,m}^{\prime }$, $b_{n,m}$, and $b_{n,m}^{\prime }$, respectively, except that for $n=0$ we count $x=0$ as the first zero of ${𝗃}_0^{\prime }\left(x\right)$. … ►

${𝗃}_n^{\prime }\left(a_{n,m}\right)=\sqrt{{\displaystyle \frac{\pi }{2j_{n+\frac{1}{2},m}}}}{J}_{n+\frac{1}{2}}^{\prime }\left(j_{n+\frac{1}{2},m}\right),$

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${𝗒}_n^{\prime }\left(b_{n,m}\right)=\sqrt{{\displaystyle \frac{\pi }{2y_{n+\frac{1}{2},m}}}}{Y}_{n+\frac{1}{2}}^{\prime }\left(y_{n+\frac{1}{2},m}\right).$

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20: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).