10 Bessel Functions Bessel and Hankel Functions 10.15 Derivatives with Respect to Order 10.17 Asymptotic Expansions for Large Argument

§10.16 Relations to Other Functions

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Notes:: For (10.16.3), (10.16.4) see Miller (1955, p. 43). For (10.16.5) and (10.16.6) see Olver (1997b, pp. 255, 259) and apply (10.27.8). For (10.16.7) and (10.16.8) apply (13.14.4) and (13.14.5). For (10.16.9) combine (10.2.2) and (16.2.1).
Referenced by:: Erratum (V1.0.18) for Paragraph Confluent Hypergeometric Functions (in §10.16)
Permalink:: http://dlmf.nist.gov/10.16
See also:: Annotations for Ch.10

Elementary Functions

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Keywords:: Bessel functions, Hankel functions, elementary functions, relations to other functions
See also:: Annotations for §10.16 and Ch.10

10.16.1	$\displaystyle J_{\frac{1}{2}}\left(z\right)$	$\displaystyle=Y_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\sin z,$
10.16.1	$\displaystyle J_{-\frac{1}{2}}\left(z\right)$	$\displaystyle=-Y_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\cos z,$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 10.1.11 and 10.1.12 (modified) Referenced by: §1.18(vi), §10.15, §10.18(iii), §10.59 Permalink: http://dlmf.nist.gov/10.16.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.16, §10.16 and Ch.10

10.16.2	$\displaystyle{H^{(1)}_{\frac{1}{2}}}\left(z\right)$	$\displaystyle=-i{H^{(1)}_{-\frac{1}{2}}}\left(z\right)=-i\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{iz},$
10.16.2	$\displaystyle{H^{(2)}_{\frac{1}{2}}}\left(z\right)$	$\displaystyle=i{H^{(2)}_{-\frac{1}{2}}}\left(z\right)=i\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{-iz}.$
ⓘ 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), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.16.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.16, §10.16 and Ch.10

For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(i).

Airy Functions

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See also:: Annotations for §10.16 and Ch.10

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

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Keywords:: Bessel functions, parabolic cylinder functions, relations to other functions
See also:: Annotations for §10.16 and Ch.10

With the notation of §12.14(i),

10.16.3	$\displaystyle J_{\frac{1}{4}}\left(z\right)$	$\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(% 0,2z^{\frac{1}{2}}\right)-W\left(0,-2z^{\frac{1}{2}}\right)\right),$
10.16.3	$\displaystyle J_{-\frac{1}{4}}\left(z\right)$	$\displaystyle=2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0% ,2z^{\frac{1}{2}}\right)+W\left(0,-2z^{\frac{1}{2}}\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $W\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function and $z$ : complex variable Referenced by: §10.16 Permalink: http://dlmf.nist.gov/10.16.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.16, §10.16 and Ch.10

10.16.4	$\displaystyle J_{\frac{3}{4}}\left(z\right)$	$\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left% (0,2z^{\frac{1}{2}}\right)-W'\left(0,-2z^{\frac{1}{2}}\right)\right),$
10.16.4	$\displaystyle J_{-\frac{3}{4}}\left(z\right)$	$\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left% (0,2z^{\frac{1}{2}}\right)+W'\left(0,-2z^{\frac{1}{2}}\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $W\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function and $z$ : complex variable Referenced by: §10.16 Permalink: http://dlmf.nist.gov/10.16.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.16, §10.16 and Ch.10

Principal values on each side of these equations correspond.

Confluent Hypergeometric Functions

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Keywords:: Bessel and Hankel functions, Bessel functions, Hankel functions, confluent hypergeometric functions, relations to other functions
Referenced by:: Erratum (V1.0.18) for Paragraph Confluent Hypergeometric Functions (in §10.16)
Clarification (effective with 1.0.18):: The confluent hypergeometric Whittaker functions in (10.16.7), (10.16.8), and the following sentence were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct.
See also:: Annotations for §10.16 and Ch.10

10.16.5		$J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{\mp iz}}{\Gamma\left(\nu+1% \right)}M\left(\nu+\tfrac{1}{2},2\nu+1,\pm 2iz\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.69 (modified) Referenced by: §10.16, §10.39 Permalink: http://dlmf.nist.gov/10.16.E5 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

10.16.6		$\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=\mp 2\pi^{-\frac{1}{2}}ie^{\mp\nu\pi i}(2z)^{% \nu}\*e^{\pm iz}U\left(\nu+\tfrac{1}{2},2\nu+1,\mp 2iz\right).$
ⓘ 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), $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.16, §10.17(v) Permalink: http://dlmf.nist.gov/10.16.E6 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

For the functions $M$ and $U$ see §13.2(i).

10.16.7		$J_{\nu}\left(z\right)=\frac{e^{\mp(2\nu+1)\pi i/4}}{2^{2\nu}\Gamma\left(\nu+1% \right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(\pm 2iz\right),$
$2\nu\neq-1,-2,-3,\dotsc$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.16, §10.16 Permalink: http://dlmf.nist.gov/10.16.E7 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

10.16.8		$\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=e^{\mp(2\nu+1)\pi i/4}\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}W_{0,\nu}\left(\mp 2iz\right).$
ⓘ 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), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.16, §10.16 Permalink: http://dlmf.nist.gov/10.16.E8 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

For the functions $M_{0,\nu}$ and $W_{0,\nu}$ see §13.14(i).

In all cases principal branches correspond at least when $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$ .

Generalized Hypergeometric Functions

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Keywords:: Bessel functions, generalized hypergeometric functions, relations to other functions
See also:: Annotations for §10.16 and Ch.10

10.16.9		$J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{% }_{0}F_{1}}\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.16 Permalink: http://dlmf.nist.gov/10.16.E9 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

For ${{}_{0}F_{1}}$ see (16.2.1).

With $\mathbf{F}$ as in §15.2(i), and with $z$ and $\nu$ fixed,

10.16.10		$J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathbf{F}\left(\lambda,\mu;\nu% +1;-z^{2}/(4\lambda\mu)\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.70 (modified) Referenced by: §10.39 Permalink: http://dlmf.nist.gov/10.16.E10 Encodings: TeX, pMML, png See also: Annotations for §10.16, §10.16 and Ch.10

as $\lambda$ and $\mu\to\infty$ in $\mathbb{C}$ . For this result see Watson (1944, §5.7).

10.15 Derivatives with Respect to Order 10.17 Asymptotic Expansions for Large Argument

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