10 Bessel FunctionsModified Bessel Functions10.29 Recurrence Relations and Derivatives10.31 Power Series

§10.30 Limiting Forms

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Referenced by:

§10.25(iii), §10.28

Permalink:

http://dlmf.nist.gov/10.30

See also:

Annotations for Ch.10

Contents

1. §10.30(i) $z\to 0$
2. §10.30(ii) $z\to \mathrm{\infty }$

§10.30(i) $z\to 0$

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Keywords:

limiting forms, modified Bessel functions

Notes:

For (10.30.1) use (10.25.2). For (10.30.2) and (10.30.3) use (10.27.4) when $\nu$ is not an integer; (10.27.3), (10.31.1) otherwise.

Referenced by:

§11.13(iv)

Permalink:

http://dlmf.nist.gov/10.30.i

See also:

Annotations for §10.30 and Ch.10

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).

§10.30(ii) $z\to \mathrm{\infty }$

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Notes:

Use (10.40.1) and (10.34.1) with $m=\pm 1$.

Referenced by:

§11.13(iv), item 3

Permalink:

http://dlmf.nist.gov/10.30.ii

See also:

Annotations for §10.30 and Ch.10

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).

10.29 Recurrence Relations and Derivatives10.31 Power Series

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