# §10.16 Relations to Other Functions

## Elementary Functions

 10.16.1 $\displaystyle J_{\frac{1}{2}}\left(z\right)$ $\displaystyle=Y_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\sin z,$ $\displaystyle J_{-\frac{1}{2}}\left(z\right)$ $\displaystyle=-Y_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\cos z,$ ⓘ 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 $\displaystyle{H^{(1)}_{\frac{1}{2}}}\left(z\right)$ $\displaystyle=-i{H^{(1)}_{-\frac{1}{2}}}\left(z\right)=-i\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{iz},$ $\displaystyle{H^{(2)}_{\frac{1}{2}}}\left(z\right)$ $\displaystyle=i{H^{(2)}_{-\frac{1}{2}}}\left(z\right)=i\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{-iz}.$

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

## Airy Functions

See §§9.6(i) and 9.6(ii).

## Parabolic Cylinder Functions

With the notation of §12.14(i),

 10.16.3 $\displaystyle J_{\frac{1}{4}}\left(z\right)$ $\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(% 0,2z^{\frac{1}{2}}\right)-W\left(0,-2z^{\frac{1}{2}}\right)\right),$ $\displaystyle J_{-\frac{1}{4}}\left(z\right)$ $\displaystyle=2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0% ,2z^{\frac{1}{2}}\right)+W\left(0,-2z^{\frac{1}{2}}\right)\right).$
 10.16.4 $\displaystyle J_{\frac{3}{4}}\left(z\right)$ $\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left% (0,2z^{\frac{1}{2}}\right)-W'\left(0,-2z^{\frac{1}{2}}\right)\right),$ $\displaystyle J_{-\frac{3}{4}}\left(z\right)$ $\displaystyle=-2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left% (0,2z^{\frac{1}{2}}\right)+W'\left(0,-2z^{\frac{1}{2}}\right)\right).$

Principal values on each side of these equations correspond.

## Confluent Hypergeometric Functions

 10.16.5 $J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{\mp iz}}{\Gamma\left(\nu+1% \right)}M\left(\nu+\tfrac{1}{2},2\nu+1,\pm 2iz\right),$
 10.16.6 $\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=\mp 2\pi^{-\frac{1}{2}}ie^{\mp\nu\pi i}(2z)^{% \nu}\*e^{\pm iz}U\left(\nu+\tfrac{1}{2},2\nu+1,\mp 2iz\right).$

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

 10.16.7 $J_{\nu}\left(z\right)=\frac{e^{\mp(2\nu+1)\pi i/4}}{2^{2\nu}\Gamma\left(\nu+1% \right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(\pm 2iz\right),$ $2\nu\neq-1,-2,-3,\dotsc$,
 10.16.8 $\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=e^{\mp(2\nu+1)\pi i/4}\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}W_{0,\nu}\left(\mp 2iz\right).$

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

 10.16.9 $J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{% }_{0}F_{1}}\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right).$

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 $J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathbf{F}\left(\lambda,\mu;\nu% +1;-z^{2}/(4\lambda\mu)\right),$

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