§10.39 Relations to Other Functions

Elementary Functions

 10.39.1 $\displaystyle I_{\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh z,$ $\displaystyle I_{-\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\cosh z,$
 10.39.2 $K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)=\left(\frac{\pi}{% 2z}\right)^{\frac{1}{2}}e^{-z}.$

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

Airy Functions

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

With the notation of §12.2(i),

 10.39.3 $K_{\frac{1}{4}}\left(z\right)=\pi^{\frac{1}{2}}z^{-\frac{1}{4}}U\left(0,2z^{% \frac{1}{2}}\right),$
 10.39.4 $K_{\frac{3}{4}}\left(z\right)=\tfrac{1}{2}\pi^{\frac{1}{2}}z^{-\frac{3}{4}}% \left(\tfrac{1}{2}U\left(1,2z^{\frac{1}{2}}\right)+U\left(-1,2z^{\frac{1}{2}}% \right)\right).$

Principal values on each side of these equations correspond. For these and further results see Miller (1955, pp. 42–43 and 77–79).

Confluent Hypergeometric Functions

 10.39.5 $\displaystyle I_{\nu}\left(z\right)$ $\displaystyle=\frac{(\tfrac{1}{2}z)^{\nu}e^{\pm z}}{\Gamma\left(\nu+1\right)}M% \left(\nu+\tfrac{1}{2},2\nu+1,\mp 2z\right),$ 10.39.6 $\displaystyle K_{\nu}\left(z\right)$ $\displaystyle=\pi^{\frac{1}{2}}(2z)^{\nu}e^{-z}U\left(\nu+\tfrac{1}{2},2\nu+1,% 2z\right),$
 10.39.7 $I_{\nu}\left(z\right)=\frac{(2z)^{-\frac{1}{2}}M_{0,\nu}\left(2z\right)}{2^{2% \nu}\Gamma\left(\nu+1\right)},$ $2\nu\neq-1,-2,-3,\dots$,
 10.39.8 $K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}W_{0,\nu}\left(% 2z\right).$

For the functions $M$, $U$, $M_{0,\nu}$, and $W_{0,\nu}$ see §§13.2(i) and 13.14(i).

Generalized Hypergeometric Functions and Hypergeometric Function

 10.39.9 $I_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+1\right)}{{}% _{0}F_{1}}\left(-;\nu+1;\tfrac{1}{4}z^{2}\right),$
 10.39.10 $I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathbf{F}\left(\lambda,\mu;\nu% +1;z^{2}/(4\lambda\mu)\right),$

as $\lambda$ and $\mu\to\infty$ in $\mathbb{C}$, with $z$ and $\nu$ fixed. For the functions ${{}_{0}F_{1}}$ and $\mathbf{F}$ see (16.2.1) and §15.2(i).