Bessel functions

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31: 10.26 Graphics

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§10.26(i) Real Order and Variable

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.9: ${\tilde{I}}_5\left(x\right)$, ${\tilde{K}}_5\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.10: ${\tilde{K}}_5\left(x\right)$, $0.01\le x\le 3$. Magnify

32: 10.10 Continued Fractions

§10.10 Continued Fractions

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10.10.1 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}-}\frac{1}{2(\nu +1)z^{-1}-}\frac{1}{2(\nu +2)z^{-1}-}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E1

Encodings:

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See also:

Annotations for §10.10 and Ch.10

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10.10.2 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1-}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1-}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1-}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E2

Encodings:

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See also:

Annotations for §10.10 and Ch.10

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33: 10.33 Continued Fractions

§10.33 Continued Fractions

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10.33.1 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}+}\frac{1}{2(\nu +1)z^{-1}+}\frac{1}{2(\nu +2)z^{-1}+}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E1

Encodings:

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See also:

Annotations for §10.33 and Ch.10

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10.33.2 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1+}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1+}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1+}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E2

Encodings:

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See also:

Annotations for §10.33 and Ch.10

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34: 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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10.49.18 $${𝗃}_n^2\left(z\right)+{𝗒}_n^2\left(z\right)=\sum _{k=0}^n\frac{s_k(n+\frac{1}{2})}{z^{2k+2}}.$$

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Symbols:

${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $s_k(n)$

A&S Ref:

10.1.27

Referenced by:

§10.49(iv)

Permalink:

http://dlmf.nist.gov/10.49.E18

Encodings:

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See also:

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

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35: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

36: 29 Lamé Functions

Chapter 29 Lamé Functions

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37: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

… ►For the functions $J$, $Y$, $I$, and $K$ see §§10.2(ii), 10.25(ii). … ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and … ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

38: 33.9 Expansions in Series of Bessel Functions

§33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

… ►where the function $𝗃$ is as in §10.47(ii), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and … ►

§33.9(ii) Bessel Functions and Modified Bessel Functions

►In this subsection the functions $J$, $I$, and $K$ are as in §§10.2(ii) and 10.25(ii). …

39: 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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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:

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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:

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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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40: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

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10.35.1 $${\mathrm{e}}^{\frac{1}{2}z(t+t^{-1})}=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{t}^m{I}_m\left(z\right).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $m$: integer and $z$: complex variable

A&S Ref:

9.6.33

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E1

Encodings:

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Annotations for §10.35 and Ch.10

►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

10.35.2 $${\mathrm{e}}^{z\cos \theta }=I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}{I}_k\left(z\right)\cos \left(k\theta \right),$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.34

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E2

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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10.35.4 $$1=I_0\left(z\right)-2I_2\left(z\right)+2I_4\left(z\right)-2I_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.36

Permalink:

http://dlmf.nist.gov/10.35.E4

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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