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

ⓘ

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

ⓘ

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

10.39.1

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

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

ⓘ

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 : 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 : 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

ⓘ

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

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

ⓘ

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 : 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 : 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

ⓘ

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

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

ⓘ

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, : 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 $K_{\nu}\left(z\right)=\pi^{\frac{1}{2}}(2z)^{\nu}e^{-z}U\left(\nu+\tfrac{1}{2}% ,2\nu+1,2z\right),$

ⓘ

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, : 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, : 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, : 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 , , $M_{0,\nu}$ , and $W_{0,\nu}$ see §§13.2(i) and 13.14(i).

Generalized Hypergeometric Functions and Hypergeometric Function

ⓘ

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, : 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, : 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 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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov