# §10.30 Limiting Forms

## §10.30(i) $z\to 0$

When $\nu$ is fixed and $z\to 0$,

 10.30.1 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\mathop{% \Gamma\/}\nolimits\!\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$,
 10.30.2 $\displaystyle\mathop{K_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim\tfrac{1}{2}\mathop{\Gamma\/}\nolimits\!\left(\nu\right)(% \tfrac{1}{2}z)^{-\nu},$ $\realpart{\nu}>0$, 10.30.3 $\displaystyle\mathop{K_{0}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim-\mathop{\ln\/}\nolimits z.$

For $\mathop{K_{\nu}\/}\nolimits\!\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7).

## §10.30(ii) $z\to\infty$

When $\nu$ is fixed and $z\to\infty$,

 10.30.4 $\displaystyle\mathop{I_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim e^{z}/\sqrt{2\pi z},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$, 10.30.5 $\displaystyle\mathop{I_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle\sim e^{\pm(\nu+\frac{1}{2})\pi i}e^{-z}/\sqrt{2\pi z},$ $\tfrac{1}{2}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}% \pi-\delta$.

For $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)$ see (10.25.3).