# §10.40 Asymptotic Expansions for Large Argument

## §10.40(i) Hankel’s Expansions

With the notation of §§10.17(i) and 10.17(ii), as $z\to\infty$ with $\nu$ fixed,

 10.40.1 $\displaystyle I_{\nu}\left(z\right)$ $\displaystyle\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{\infty}(-1)^{% k}\frac{a_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$, 10.40.2 $\displaystyle K_{\nu}\left(z\right)$ $\displaystyle\sim\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum_{k=0}^{% \infty}\frac{a_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$,
 10.40.3 $I_{\nu}'\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$,
 10.40.4 $K_{\nu}'\left(z\right)\sim-\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum% _{k=0}^{\infty}\frac{b_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.

Corresponding expansions for $I_{\nu}\left(z\right)$, $K_{\nu}\left(z\right)$, $I_{\nu}'\left(z\right)$, and $K_{\nu}'\left(z\right)$ for other ranges of $\operatorname{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with $m=0$ yields the following more general (and more accurate) version of (10.40.1):

 10.40.5 $I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu\pi i}\frac{e^{-z}}{(2\pi z% )^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$ $-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$.

### Products

With $\mu=4\nu^{2}$ and fixed,

 10.40.6 $\displaystyle I_{\nu}\left(z\right)K_{\nu}\left(z\right)$ $\displaystyle\sim\frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu-1}{(2z)^{2}}+\frac{1% \cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}}-\cdots\right),$ 10.40.7 $\displaystyle I_{\nu}'\left(z\right)K_{\nu}'\left(z\right)$ $\displaystyle\sim-\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu-3}{(2z)^{2}}-\frac{% 1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+\cdots\right),$

as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$. The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20).

### $\nu$-Derivative

For fixed $\nu$,

 10.40.8 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\sim\left(\frac{\pi}{2z}% \right)^{\frac{1}{2}}\frac{\nu e^{-z}}{z}\sum_{k=0}^{\infty}\frac{\alpha_{k}(% \nu)}{(8z)^{k}},$

as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$. Here $\alpha_{0}(\nu)=1$ and

 10.40.9 $\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{% 2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+% \frac{1}{4\nu^{2}-(2k+1)^{2}}\right).$ ⓘ Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.40.E9 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

## §10.40(ii) Error Bounds for Real Argument and Order

In the expansion (10.40.2) assume that $z>0$ and the sum is truncated when $k=\ell-1$. Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that $\ell\geq\max(|\nu|-\tfrac{1}{2},1)$.

For the error term in (10.40.1) see §10.40(iii).

## §10.40(iii) Error Bounds for Complex Argument and Order

For (10.40.2) write

 10.40.10 $K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\left(% \sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}(\nu,z)\right),$ $\ell=1,2,\ldots$.

Then

 10.40.11 $|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),$ ⓘ Defines: $R_{\ell}(\nu,z)$: remainder (locally) Symbols: $\exp\NVar{z}$: exponential function, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_{k}(\nu)$: expansion Permalink: http://dlmf.nist.gov/10.40.E11 Encodings: TeX, pMML, png See also: Annotations for §10.40(iii), §10.40 and Ch.10

where $\mathcal{V}$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\Re t|$ changes monotonically. Bounds for $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)$ are given by

 10.40.12 $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$: phase, $\Re$: real part, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation and $z$: complex variable Referenced by: §10.40(i), §10.40(iii), Erratum Equation (10.40.12) Permalink: http://dlmf.nist.gov/10.40.E12 Encodings: TeX, pMML, png Correction (effective with 1.0.11): Originally the third constraint $\pi\leq|\operatorname{ph}z|<\frac{3}{2}\pi$ was written incorrectly as $\pi\leq|\operatorname{ph}z|\leq\frac{3}{2}\pi$. Suggested 2014-11-05 by Gergő Nemes See also: Annotations for §10.40(iii), §10.40 and Ch.10

where $\chi(\ell)=\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}\ell+1\right)/\Gamma\left(% \tfrac{1}{2}\ell+\tfrac{1}{2}\right)$; see §9.7(i).

A similar result for (10.40.1) is obtained by combining (10.34.3), with $m=0$, and (10.40.10)–(10.40.12); see Olver (1997b, p. 269).

## §10.40(iv) Exponentially-Improved Expansions

In (10.40.10)

 10.40.13 $R_{\ell}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m-1}% \frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(2z\right)+R_{m,\ell}(\nu,z)\right),$

where $G_{p}\left(z\right)$ is given by (10.17.16). If $z\to\infty$ with $|\ell-2|z||$ bounded and $m$ $(\geq 0)$ fixed, then

 10.40.14 $R_{m,\ell}(\nu,z)=O\left(e^{-2|z|}z^{-m}\right),$ $|\operatorname{ph}z|\leq\pi$. ⓘ Defines: $R_{m,\ell}(\nu,z)$: remainder (locally) Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\operatorname{ph}$: phase, $m$: integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.40.E14 Encodings: TeX, pMML, png See also: Annotations for §10.40(iv), §10.40 and Ch.10

For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).