§10.30 Limiting Forms

When $\nu$ is fixed and $z\to 0$,

10.30.1		$$I_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$
$\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.7 Referenced by: §10.30(i) Permalink: http://dlmf.nist.gov/10.30.E1 Encodings: TeX, pMML, png See also: Annotations for §10.30(i), §10.30 and Ch.10

10.30.2		$K_{\nu }\left(z\right)$	$\sim \frac{1}{2}\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$
$\mathrm{\Re }\nu >0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.9 Referenced by: §10.30(i) Permalink: http://dlmf.nist.gov/10.30.E2 Encodings: TeX, pMML, png See also: Annotations for §10.30(i), §10.30 and Ch.10
10.30.3		$K_0\left(z\right)$	$\sim -\ln z.$
ⓘ Symbols: $\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable A&S Ref: 9.6.8 Referenced by: §10.30(i) Permalink: http://dlmf.nist.gov/10.30.E3 Encodings: TeX, pMML, png See also: Annotations for §10.30(i), §10.30 and Ch.10

For $K_{\nu }\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7).

When $\nu$ is fixed and $z\to \mathrm{\infty }$,

10.30.4		$I_{\nu }\left(z\right)$	$\sim {\mathrm{e}}^z/\sqrt{2\pi z},$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant Permalink: http://dlmf.nist.gov/10.30.E4 Encodings: TeX, pMML, png See also: Annotations for §10.30(ii), §10.30 and Ch.10
10.30.5		$I_{\nu }\left(z\right)$	$\sim {\mathrm{e}}^{\pm (\nu +\frac{1}{2})\pi \mathrm{i}}{\mathrm{e}}^{-z}/\sqrt{2\pi z},$
$\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.
ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant Permalink: http://dlmf.nist.gov/10.30.E5 Encodings: TeX, pMML, png See also: Annotations for §10.30(ii), §10.30 and Ch.10

For $K_{\nu }\left(z\right)$ see (10.25.3).