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DLMF: §10.30 Limiting Forms ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Modified Bessel Functions 10.29 Recurrence Relations and Derivatives 10.31 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

§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		$I_{\nu}\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\Gamma\left(\nu+1\right),$
$\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.7 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		$\displaystyle K_{\nu}\left(z\right)$	$\displaystyle\sim\tfrac{1}{2}\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$
$\Re\nu>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : 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		$\displaystyle K_{0}\left(z\right)$	$\displaystyle\sim-\ln z.$
ⓘ Symbols: $\sim$ : asymptotic equality, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : 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) $z\to\infty$

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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 $z\to\infty$ ,

10.30.4		$\displaystyle I_{\nu}\left(z\right)$	$\displaystyle\sim e^{z}/\sqrt{2\pi z},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : 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		$\displaystyle I_{\nu}\left(z\right)$	$\displaystyle\sim e^{\pm(\nu+\frac{1}{2})\pi i}e^{-z}/\sqrt{2\pi z},$
$\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$ .
ⓘ 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, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : 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 Derivatives 10.31 Power Series

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