§10.39 Relations to Other Functions

Notes:: For (10.39.5)–(10.39.10) combine (10.16.5)–(10.16.10) with (10.27.6) and (10.27.8).
Permalink:: http://dlmf.nist.gov/10.39

¶ Elementary Functions

Keywords:: elementary functions, modified Bessel functions

10.39.1

$\mathop{I_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\sinh\/}\nolimits z,$

$\mathop{I_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\cosh\/}\nolimits z,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.39.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

10.39.2 $\mathop{K_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{K_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{\pi}{2z}\right)^{{\frac{1}{2}}}e^{{-z}}.$

Symbols:: $e$ : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $z$ : complex variable
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/10.39.E2
Encodings:: TeX, pMML, png

For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(ii).

¶ Airy Functions

See §§9.6(i) and 9.6(ii).

¶ Parabolic Cylinder Functions

Keywords:: modified Bessel functions, parabolic cylinder functions

With the notation of §12.2(i),

10.39.3 $\mathop{K_{{\frac{1}{4}}}\/}\nolimits\!\left(z\right)=\pi^{{\frac{1}{2}}}z^{{-\frac{1}{4}}}\mathop{U\/}\nolimits\!\left(0,2z^{{\frac{1}{2}}}\right),$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.39.E3
Encodings:: TeX, pMML, png

10.39.4 $\mathop{K_{{\frac{3}{4}}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi^{{\frac{1}{2}}}z^{{-\frac{3}{4}}}\left(\tfrac{1}{2}\mathop{U\/}\nolimits\!\left(1,2z^{{\frac{1}{2}}}\right)+\mathop{U\/}\nolimits\!\left(-1,2z^{{\frac{1}{2}}}\right)\right).$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.39.E4
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Principal values on each side of these equations correspond. For these and further results see Miller (1955, pp. 42–43 and 77–79).

¶ Confluent Hypergeometric Functions

Keywords:: confluent hypergeometric functions, modified Bessel functions

10.39.5 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{{\pm z}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}\mathop{M\/}\nolimits\!\left(\nu+\tfrac{1}{2},2\nu+1,\mp 2z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.39
Permalink:: http://dlmf.nist.gov/10.39.E5
Encodings:: TeX, pMML, png

10.39.6 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\pi^{{\frac{1}{2}}}(2z)^{\nu}e^{{-z}}\mathop{U\/}\nolimits\!\left(\nu+\tfrac{1}{2},2\nu+1,2z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.40(iv)
Permalink:: http://dlmf.nist.gov/10.39.E6
Encodings:: TeX, pMML, png

10.39.7 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(2z)^{{-\frac{1}{2}}}\mathop{M_{{0,\nu}}\/}\nolimits\!\left(2z\right)}{2^{{2\nu}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)},$ $2\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.47
Permalink:: http://dlmf.nist.gov/10.39.E7
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10.39.8 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\left(\frac{\pi}{2z}\right)^{{\frac{1}{2}}}\mathop{W_{{0,\nu}}\/}\nolimits\!\left(2z\right).$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.48
Referenced by:: §9.6(iii)
Permalink:: http://dlmf.nist.gov/10.39.E8
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For the functions $\mathop{M\/}\nolimits$ , $\mathop{U\/}\nolimits$ , $\mathop{M_{{0,\nu}}\/}\nolimits$ , and $\mathop{W_{{0,\nu}}\/}\nolimits$ see §§13.2(i) and 13.14(i).

¶ Generalized Hypergeometric Functions and Hypergeometric Function

Keywords:: generalized hypergeometric functions, modified Bessel functions

10.39.9 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits\!\left(-;\nu+1;\tfrac{1}{4}z^{2}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.39.E9
Encodings:: TeX, pMML, png

10.39.10 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathop{{{\mathbf{F}}}\/}\nolimits\!\left(\lambda,\mu;\nu+1;z^{2}/(4\lambda\mu)\right),$

Symbols:: $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.39, §9.6(iii)
Permalink:: http://dlmf.nist.gov/10.39.E10
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as $\lambda$ and $\mu\to\infty$ in $\Complex$ , with $z$ and $\nu$ fixed. For the functions $\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits$ and $\mathop{{{\mathbf{F}}}\/}\nolimits$ see (16.2.1) and §15.2(i).