§10.39 Relations to Other Functions

Elementary Functions

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

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

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

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

Generalized Hypergeometric Functions and Hypergeometric Function

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

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