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

§10.7(i) $z\to 0$
§10.7(ii) $z\to\infty$

§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

$J_{0}\left(z\right)\to 1,$

$Y_{0}\left(z\right)\sim(2/\pi)\ln z,$

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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 : 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 ${H^{(1)}_{0}}\left(z\right)\sim-{H^{(2)}_{0}}\left(z\right)\sim(2i/\pi)\ln z,$

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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 : 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 $J_{\nu}\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\Gamma\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, : 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 $Y_{\nu}\left(z\right)\sim-(1/\pi)\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$ $\Re\nu>0$ or $\nu=-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\dotsc$ ,

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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, : 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 $Y_{-\nu}\left(z\right)\sim-(1/\pi)\cos\left(\nu\pi\right)\Gamma\left(\nu\right% )(\tfrac{1}{2}z)^{-\nu},$ $\Re\nu>0$ , $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dotsc$ ,

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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, : 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 $\nu\neq 0$ .

§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

$J_{\nu}\left(z\right)=\sqrt{2/(\pi z)}\left(\cos\left(z-\tfrac{1}{2}\nu\pi-% \tfrac{1}{4}\pi\right)+e^{|\Im z|}o\left(1\right)\right),$

$Y_{\nu}\left(z\right)=\sqrt{2/(\pi z)}\left(\sin\left(z-\tfrac{1}{2}\nu\pi-% \tfrac{1}{4}\pi\right)+e^{|\Im z|}o\left(1\right)\right),$ $|\operatorname{ph}z|\leq\pi-\delta(<\pi)$ .