§10.39 Relations to Other Functions

10.39.1		$I_{\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\sinh z,$
		$I_{-\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\cosh z,$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $I_{\nu }\left(z\right)$: 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		$$K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}.$$
ⓘ Symbols: $\pi$: 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 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).

See §§9.6(i) and 9.6(ii).

With the notation of §12.2(i),

10.39.3		$$K_{\frac{1}{4}}\left(z\right)={\pi }^{\frac{1}{2}}{z}^{-\frac{1}{4}}U(0,2z^{\frac{1}{2}}),$$
ⓘ 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 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		$$K_{\frac{3}{4}}\left(z\right)=\frac{1}{2}{\pi }^{\frac{1}{2}}{z}^{-\frac{3}{4}}\left(\frac{1}{2}U(1,2z^{\frac{1}{2}})+U(-1,2z^{\frac{1}{2}})\right).$$
ⓘ 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 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).

10.39.5		$I_{\nu }\left(z\right)$	$={\displaystyle \frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\pm z}}{\mathrm{\Gamma }\left(\nu +1\right)}}M(\nu +\frac{1}{2},2\nu +1,\mp 2z),$
ⓘ 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, $z$: 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		$K_{\nu }\left(z\right)$	$={\pi }^{\frac{1}{2}}{(2z)}^{\nu }{\mathrm{e}}^{-z}U(\nu +\frac{1}{2},2\nu +1,2z),$
ⓘ 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, $z$: 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		$$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$
$2\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ 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, $z$: 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: $W_{\kappa ,\mu }\left(z\right)$: 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, $z$: 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).

10.39.9		$$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}{}_{0}F_1(-;\nu +1;\frac{1}{4}{z}^2),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $I_{\nu }\left(z\right)$: 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		$$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim\mathbf{F}(\lambda ,\mu ;\nu +1;z^2/(4\lambda \mu )),$$
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mu \to \mathrm{\infty }$ in $ℂ$, with $z$ and $\nu$ fixed. For the functions ${}_{0}F_1$ and $\mathbf{F}$ see (16.2.1) and §15.2(i).