10 Bessel Functions Modified Bessel Functions 10.38 Derivatives with Respect to Order 10.40 Asymptotic Expansions for Large Argument

§10.39 Relations to Other Functions

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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
See also:: Annotations for Ch.10

Elementary Functions

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Keywords:: elementary functions, modified Bessel functions, relations to other functions
See also:: Annotations for §10.39 and Ch.10

10.39.1	$\displaystyle I_{\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh z,$
10.39.1	$\displaystyle I_{-\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\cosh z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.39.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.39, §10.39 and Ch.10

10.39.2		$K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)=\left(\frac{\pi}{% 2z}\right)^{\frac{1}{2}}e^{-z}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $z$ : complex variable Referenced by: §7.14(i) Permalink: http://dlmf.nist.gov/10.39.E2 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

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

Airy Functions

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See also:: Annotations for §10.39 and Ch.10

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

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Keywords:: modified Bessel functions, parabolic cylinder functions, relations to other functions
See also:: Annotations for §10.39 and Ch.10

With the notation of §12.2(i),

10.39.3		$K_{\frac{1}{4}}\left(z\right)=\pi^{\frac{1}{2}}z^{-\frac{1}{4}}U\left(0,2z^{% \frac{1}{2}}\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.39.E3 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

10.39.4		$K_{\frac{3}{4}}\left(z\right)=\tfrac{1}{2}\pi^{\frac{1}{2}}z^{-\frac{3}{4}}% \left(\tfrac{1}{2}U\left(1,2z^{\frac{1}{2}}\right)+U\left(-1,2z^{\frac{1}{2}}% \right)\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.39.E4 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

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

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Keywords:: confluent hypergeometric functions, modified Bessel functions, relations to other functions
See also:: Annotations for §10.39 and Ch.10

10.39.5	$\displaystyle I_{\nu}\left(z\right)$	$\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu}e^{\pm z}}{\Gamma\left(\nu+1\right)}M% \left(\nu+\tfrac{1}{2},2\nu+1,\mp 2z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.39 Permalink: http://dlmf.nist.gov/10.39.E5 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10
10.39.6	$\displaystyle K_{\nu}\left(z\right)$	$\displaystyle=\pi^{\frac{1}{2}}(2z)^{\nu}e^{-z}U\left(\nu+\tfrac{1}{2},2\nu+1,% 2z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.40(iv), (9.6.21), (9.6.22) Permalink: http://dlmf.nist.gov/10.39.E6 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

10.39.7		$I_{\nu}\left(z\right)=\frac{(2z)^{-\frac{1}{2}}M_{0,\nu}\left(2z\right)}{2^{2% \nu}\Gamma\left(\nu+1\right)},$
$2\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.47 Referenced by: (9.6.23), (9.6.24) Permalink: http://dlmf.nist.gov/10.39.E7 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

10.39.8		$K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}W_{0,\nu}\left(% 2z\right).$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.48 Referenced by: (9.6.21), (9.6.22) Permalink: http://dlmf.nist.gov/10.39.E8 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

For the functions $M$ , $U$ , $M_{0,\nu}$ , and $W_{0,\nu}$ see §§13.2(i) and 13.14(i).

Generalized Hypergeometric Functions and Hypergeometric Function

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Keywords:: generalized hypergeometric functions, modified Bessel functions, relations to other functions
See also:: Annotations for §10.39 and Ch.10

10.39.9		$I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{}% _{0}F_{1}}\left(-;\nu+1;\tfrac{1}{4}z^{2}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §9.6(iii) Permalink: http://dlmf.nist.gov/10.39.E9 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

10.39.10		$I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathbf{F}\left(\lambda,\mu;\nu% +1;z^{2}/(4\lambda\mu)\right),$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.39 Permalink: http://dlmf.nist.gov/10.39.E10 Encodings: TeX, pMML, png See also: Annotations for §10.39, §10.39 and Ch.10

as $\lambda$ and $\mu\to\infty$ in $\mathbb{C}$ , with $z$ and $\nu$ fixed. For the functions ${{}_{0}F_{1}}$ and $\mathbf{F}$ see (16.2.1) and §15.2(i).

10.38 Derivatives with Respect to Order 10.40 Asymptotic Expansions for Large Argument

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