modified Bessel functions

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31: 10.36 Other Differential Equations

§10.36 Other Differential Equations

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32: 13.18 Relations to Other Functions

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§13.18(iii) Modified Bessel Functions

►When $\kappa =0$ the Whittaker functions can be expressed as modified Bessel functions. … ►

13.18.8 $$M_{0,\nu }\left(2z\right)=2^{2\nu +\frac{1}{2}}\mathrm{\Gamma }\left(1+\nu \right)\sqrt{z}{I}_{\nu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

Referenced by:

§13.24(ii)

Permalink:

http://dlmf.nist.gov/13.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iii), §13.18 and Ch.13

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13.18.9 $$W_{0,\nu }\left(2z\right)=\sqrt{2z/\pi }{K}_{\nu }\left(z\right),$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable

Referenced by:

(18.34.2)

Permalink:

http://dlmf.nist.gov/13.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iii), §13.18 and Ch.13

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33: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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$$F_0(\eta ,\rho )\sim {\mathrm{e}}^{-\pi \eta }{(\pi \rho )}^{1/2}{I}_1\left({(8\eta \rho )}^{1/2}\right),$$

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$$G_0(\eta ,\rho )\sim 2{\mathrm{e}}^{\pi \eta }{\left(\rho /\pi \right)}^{1/2}{K}_1\left({(8\eta \rho )}^{1/2}\right),$$

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§33.10(iii) Large Negative $\eta$

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34: 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). … ►

33.20.5 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(-1)}^{\mathrm{\ell }+1+p}{(2|r|)}^{(p+1)/2}{C}_{k,p}{I}_{2\mathrm{\ell }+1+p}\left(\sqrt{8|r|}\right),$$ .

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

►The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

35: 13.24 Series

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§13.24(ii) Expansions in Series of Bessel Functions

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13.24.1 $$M_{\kappa ,\mu }\left(z\right)=\mathrm{\Gamma }\left(\kappa +\mu \right)2^{2\kappa +2\mu }{z}^{\frac{1}{2}-\kappa }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2\kappa +2\mu \right)}_s{\left(2\kappa \right)}_s}{{\left(1+2\mu \right)}_{s}s!}\left(\kappa +\mu +s\right)I_{\kappa +\mu +s}\left(\frac{1}{2}z\right),$$ $2\mu ,\kappa +\mu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.11.2), §13.24(ii)

Permalink:

http://dlmf.nist.gov/13.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

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36: 13.6 Relations to Other Functions

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§13.6(iii) Modified Bessel Functions

►When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. … ►

13.6.9 $$M(\nu +\frac{1}{2},2\nu +1,2z)=\mathrm{\Gamma }\left(1+\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }{I}_{\nu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

13.6.3

Referenced by:

§13.11, §7.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.10 $$U(\nu +\frac{1}{2},2\nu +1,2z)=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^z{\left(2z\right)}^{-\nu }{K}_{\nu }\left(z\right),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable

A&S Ref:

13.6.21

Permalink:

http://dlmf.nist.gov/13.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.11_1 $$M(\nu +\frac{1}{2},2\nu +1+n,2z)=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }\sum _{k=0}^n\frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}{I}_{\nu +k}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.6))

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.6.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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37: 13.11 Series

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13.11.1 $$M(a,b,z)=\mathrm{\Gamma }\left(a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{\frac{1}{2}-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2a-1\right)}_s{\left(2a-b\right)}_s}{{\left(b\right)}_{s}s!}\left(a-\frac{1}{2}+s\right)I_{a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(iii), §13.6(iii), §8.7

Permalink:

http://dlmf.nist.gov/13.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.11 and Ch.13

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13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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$(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\mathrm{\dots }$.

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38: 12.7 Relations to Other Functions

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§12.7(iii) Modified Bessel Functions

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12.7.8 $$U(-2,z)=\frac{z^{5/2}}{4\sqrt{2\pi }}\left(2K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+3K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{5}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.9 $$U(-1,z)=\frac{z^{3/2}}{2\sqrt{2\pi }}\left(K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.10 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.10 $$U(0,z)=\sqrt{\frac{z}{2\pi }}{K}_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.9 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.11 $$U(1,z)=\frac{z^{3/2}}{\sqrt{2\pi }}\left(K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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39: 8.7 Series Expansions

§8.7 Series Expansions

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8.7.4 $$\gamma (a,x)=\mathrm{\Gamma }\left(a\right)x^{\frac{1}{2}a}{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}{e}_n(-1)x^{\frac{1}{2}n}{I}_{n+a}\left(2x^{1/2}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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8.7.5 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){𝗂}_n^{(1)}\left(\frac{1}{2}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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40: Bibliography T

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On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19 (3), pp. 324–337.

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Notes:

Algol program.

External Links:

Document, MathReview (Carl C. Grosjean), ZentralBlatt

Referenced by:

§10.74(i), §10.74(iv), §10.77(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

Encodings:

BibTeX

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

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Further comments on the computation of modified Bessel function ratios. Math. Comp. 35 (151), pp. 937–939.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (James M. Horner), ZentralBlatt

Referenced by:

§10.74(v).

Encodings:

BibTeX

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