§10.16 Relations to Other Functions

10.16.1		$J_{\frac{1}{2}}\left(z\right)$	$=Y_{-\frac{1}{2}}\left(z\right)={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\sin z,$
		$J_{-\frac{1}{2}}\left(z\right)$	$=-Y_{\frac{1}{2}}\left(z\right)={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\cos z,$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable A&S Ref: 10.1.11 and 10.1.12 (modified) Referenced by: §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		$H_{\frac{1}{2}}^{(1)}\left(z\right)$	$=-\mathrm{i}{H}_{-\frac{1}{2}}^{(1)}\left(z\right)=-\mathrm{i}{\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}z},$
		$H_{\frac{1}{2}}^{(2)}\left(z\right)$	$=\mathrm{i}{H}_{-\frac{1}{2}}^{(2)}\left(z\right)=\mathrm{i}{\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}z}.$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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).

See §§9.6(i) and 9.6(ii).

With the notation of §12.14(i),

10.16.3		$J_{\frac{1}{4}}\left(z\right)$	$=-2^{-\frac{1}{4}}{\pi }^{-\frac{1}{2}}{z}^{-\frac{1}{4}}\left(W(0,2z^{\frac{1}{2}})-W(0,-2z^{\frac{1}{2}})\right),$
		$J_{-\frac{1}{4}}\left(z\right)$	$=2^{-\frac{1}{4}}{\pi }^{-\frac{1}{2}}{z}^{-\frac{1}{4}}\left(W(0,2z^{\frac{1}{2}})+W(0,-2z^{\frac{1}{2}})\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $W(a,x)$: 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		$J_{\frac{3}{4}}\left(z\right)$	$=-2^{-\frac{1}{4}}{\pi }^{-\frac{1}{2}}{z}^{-\frac{3}{4}}\left(W^{\prime }(0,2z^{\frac{1}{2}})-W^{\prime }(0,-2z^{\frac{1}{2}})\right),$
		$J_{-\frac{3}{4}}\left(z\right)$	$=-2^{-\frac{1}{4}}{\pi }^{-\frac{1}{2}}{z}^{-\frac{3}{4}}\left(W^{\prime }(0,2z^{\frac{1}{2}})+W^{\prime }(0,-2z^{\frac{1}{2}})\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $W(a,x)$: 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.

10.16.5		$$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\mp \mathrm{i}z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\pm 2\mathrm{i}z),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ 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		$$\begin{array}{c}\hfill H_{\nu }^{(1)}\left(z\right)\hfill \\ \hfill H_{\nu }^{(2)}\left(z\right)\hfill \end{array}\}=\mp 2{\pi }^{-\frac{1}{2}}\mathrm{i}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}}{(2z)}^{\nu }{\mathrm{e}}^{\pm \mathrm{i}z}U(\nu +\frac{1}{2},2\nu +1,\mp 2\mathrm{i}z).$$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $U(a,b,z)$: 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		$$J_{\nu }\left(z\right)=\frac{{\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)}{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(\pm 2\mathrm{i}z\right),$$
$2\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: 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		$$\begin{array}{c}\hfill H_{\nu }^{(1)}\left(z\right)\hfill \\ \hfill H_{\nu }^{(2)}\left(z\right)\hfill \end{array}\}={\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(\mp 2\mathrm{i}z\right).$$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: 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 $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

10.16.9		$$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}{}_{0}F_1(-;\nu +1;-\frac{1}{4}{z}^2).$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ 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		$$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim\mathbf{F}(\lambda ,\mu ;\nu +1;-z^2/(4\lambda \mu )),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 $\mu \to \mathrm{\infty }$ in $ℂ$. For this result see Watson (1944, §5.7).