# §10.30 Limiting Forms

## §10.30(i) $z\to 0$

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

 10.30.1 $I_{\nu}\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\Gamma\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: asymptotic equality, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.7 Referenced by: §10.30(i) Permalink: http://dlmf.nist.gov/10.30.E1 Encodings: TeX, pMML, png See also: Annotations for §10.30(i), §10.30 and Ch.10
 10.30.2 $\displaystyle K_{\nu}\left(z\right)$ $\displaystyle\sim\tfrac{1}{2}\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu},$ $\Re\nu>0$, 10.30.3 $\displaystyle K_{0}\left(z\right)$ $\displaystyle\sim-\ln z.$ ⓘ Symbols: $\sim$: asymptotic equality, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 9.6.8 Referenced by: §10.30(i) Permalink: http://dlmf.nist.gov/10.30.E3 Encodings: TeX, pMML, png See also: Annotations for §10.30(i), §10.30 and Ch.10

For $K_{\nu}\left(x\right)$, 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 $\displaystyle I_{\nu}\left(z\right)$ $\displaystyle\sim e^{z}/\sqrt{2\pi z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$, 10.30.5 $\displaystyle I_{\nu}\left(z\right)$ $\displaystyle\sim e^{\pm(\nu+\frac{1}{2})\pi i}e^{-z}/\sqrt{2\pi z},$ $\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$.

For $K_{\nu}\left(z\right)$ see (10.25.3).