§10.16 Relations to Other Functions

Elementary Functions

 10.16.1 $\displaystyle\mathop{J_{\frac{1}{2}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{Y_{-\frac{1}{2}}\/}\nolimits\!\left(z\right)=\left(\frac% {2}{\pi z}\right)^{\frac{1}{2}}\mathop{\sin\/}\nolimits z,$ $\displaystyle\mathop{J_{-\frac{1}{2}}\/}\nolimits\!\left(z\right)$ $\displaystyle=-\mathop{Y_{\frac{1}{2}}\/}\nolimits\!\left(z\right)=\left(\frac% {2}{\pi z}\right)^{\frac{1}{2}}\mathop{\cos\/}\nolimits z,$ Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the second kind, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 10.16
 10.16.2 $\displaystyle\mathop{{H^{(1)}_{\frac{1}{2}}}\/}\nolimits\!\left(z\right)$ $\displaystyle=-i\mathop{{H^{(1)}_{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=-% i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz},$ $\displaystyle\mathop{{H^{(2)}_{\frac{1}{2}}}\/}\nolimits\!\left(z\right)$ $\displaystyle=i\mathop{{H^{(2)}_{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=i% \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}.$

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

Principal values on each side of these equations correspond.

Confluent Hypergeometric Functions

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

For the functions $\mathop{M\/}\nolimits$ and $\mathop{U\/}\nolimits$ see §13.2(i).

 10.16.7 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)=\frac{e^{\mp(2\nu+1)\pi i/4}}{2^{2% \nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}(2z)^{-\frac{1}{2}}\mathop{% M\/}\nolimits_{0,\nu}(\pm 2iz),$ $2\nu\neq-1,-2-3,\ldots$,
 10.16.8 $\rselection{\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)\\ \mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)}=e^{\mp(2\nu+1)\pi i/4}% \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\mathop{W\/}\nolimits_{0,\nu}(\mp 2% iz).$

For the functions $\mathop{M\/}\nolimits_{0,\nu}$ and $\mathop{W\/}\nolimits_{0,\nu}$ see §13.14(i).

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

Generalized Hypergeometric Functions

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

For $\mathop{{{}_{0}F_{1}}\/}\nolimits$ see (16.2.1).

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

 10.16.10 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathop{{% {\mathbf{F}}}\/}\nolimits\!\left(\lambda,\mu;\nu+1;-z^{2}/(4\lambda\mu)\right),$

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