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DLMF: §10.7 Limiting Forms ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Bessel and Hankel Functions 10.6 Recurrence Relations and Derivatives 10.8 Power Series

§10.7 Limiting Forms

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Keywords:: Bessel functions, Hankel functions, limiting forms
Referenced by:: §10.2(iii), §10.5
Permalink:: http://dlmf.nist.gov/10.7
See also:: Annotations for Ch.10

Contents

§10.7(i) $z\to 0$

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Notes:: For (10.7.1) and (10.7.3) use (10.2.2) and (10.8.2). For (10.7.2) use (10.4.3) and (10.7.1). For (10.7.4) and (10.7.5) use (10.2.3) and (10.7.3) when $\nu$ is not an integer; (10.4.1), (10.8.1) otherwise. For (10.7.6) use (10.2.3) and (10.7.3). For (10.7.7) use (10.4.3), (10.7.3), and (10.7.4).
Referenced by:: §10.18(i), §11.13(iv)
Permalink:: http://dlmf.nist.gov/10.7.i
See also:: Annotations for §10.7 and Ch.10

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

10.7.1		$\displaystyle J_{0}\left(z\right)$	$\displaystyle\to 1,$
10.7.1		$\displaystyle Y_{0}\left(z\right)$	$\displaystyle\sim(2/\pi)\ln z,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 9.1.8 Referenced by: §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.7(i), §10.7 and Ch.10

10.7.2		${H^{(1)}_{0}}\left(z\right)\sim-{H^{(2)}_{0}}\left(z\right)\sim(2i/\pi)\ln z,$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 9.1.8 Referenced by: §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E2 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10

10.7.3		$J_{\nu}\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\Gamma\left(\nu+1\right),$
$\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.7 Referenced by: §10.7(i), §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E3 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10

10.7.4		$\displaystyle Y_{\nu}\left(z\right)$	$\displaystyle\sim-(1/\pi)\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$
$\Re\nu>0$ or $\nu=-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\dotsc$ ,
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.9 Referenced by: §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E4 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10
10.7.5		$\displaystyle Y_{-\nu}\left(z\right)$	$\displaystyle\sim-(1/\pi)\cos\left(\nu\pi\right)\Gamma\left(\nu\right)(\tfrac{% 1}{2}z)^{-\nu},$
$\Re\nu>0$ , $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dotsc$ ,
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E5 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10

10.7.6		$Y_{i\nu}\left(z\right)=\frac{i\operatorname{csch}\left(\nu\pi\right)}{\Gamma% \left(1-i\nu\right)}(\tfrac{1}{2}z)^{-i\nu}-\frac{i\coth\left(\nu\pi\right)}{% \Gamma\left(1+i\nu\right)}(\tfrac{1}{2}z)^{i\nu}+e^{\|\nu\operatorname{ph}z\|}o% \left(1\right),$
$\nu\in\mathbb{R}$ and $\nu\neq 0$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than, $\operatorname{ph}$ : phase, $\mathbb{R}$ : real line, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.7(i), §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E6 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10

See also §10.24 when $z=x$ $(>0)$ .

10.7.7		${H^{(1)}_{\nu}}\left(z\right)\sim-{H^{(2)}_{\nu}}\left(z\right)\sim-(i/\pi)% \Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$
$\Re\nu>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.7(i), §10.7(i) Permalink: http://dlmf.nist.gov/10.7.E7 Encodings: TeX, pMML, png See also: Annotations for §10.7(i), §10.7 and Ch.10

For ${H^{(1)}_{-\nu}}\left(z\right)$ and ${H^{(2)}_{-\nu}}\left(z\right)$ when $\Re\nu>0$ combine (10.4.6) and (10.7.7). For ${H^{(1)}_{i\nu}}\left(z\right)$ and ${H^{(2)}_{i\nu}}\left(z\right)$ when $\nu\in\mathbb{R}$ and $\nu\neq 0$ combine (10.4.3), (10.7.3), and (10.7.6).

§10.7(ii) $z\to\infty$

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Notes:: See (10.17.3) and (10.17.4).
Referenced by:: §11.13(iv), item 3
Permalink:: http://dlmf.nist.gov/10.7.ii
See also:: Annotations for §10.7 and Ch.10

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

10.7.8		$\displaystyle J_{\nu}\left(z\right)$	$\displaystyle=\sqrt{2/(\pi z)}\left(\cos\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4% }\pi\right)+e^{\|\Im z\|}o\left(1\right)\right),$
		$\displaystyle Y_{\nu}\left(z\right)$	$\displaystyle=\sqrt{2/(\pi z)}\left(\sin\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4% }\pi\right)+e^{\|\Im z\|}o\left(1\right)\right),$
	$\|\operatorname{ph}z\|\leq\pi-\delta(<\pi)$ .
	ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $o\left(\NVar{x}\right)$ : order less than, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\nu$ : complex parameter and $\delta$ : small positive constant A&S Ref: 9.2.1, 9.2.2 Permalink: http://dlmf.nist.gov/10.7.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.7(ii), §10.7 and Ch.10

For the corresponding results for ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ see (10.2.5) and (10.2.6).

10.6 Recurrence Relations and Derivatives 10.8 Power Series

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