# §10.41 Asymptotic Expansions for Large Order

## §10.41(i) Asymptotic Forms

If $\nu\to\infty$ through positive real values with $z(\neq 0)$ fixed, then

 10.41.1 $I_{\nu}\left(z\right)\sim\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^% {\nu},$
 10.41.2 $K_{\nu}\left(z\right)\sim\sqrt{\frac{\pi}{2\nu}}\left(\frac{ez}{2\nu}\right)^{% -\nu}.$

## §10.41(ii) Uniform Expansions for Real Variable

As $\nu\to\infty$ through positive real values,

 10.41.3 $\displaystyle I_{\nu}\left(\nu z\right)$ $\displaystyle\sim\frac{e^{\nu\eta}}{(2\pi\nu)^{\frac{1}{2}}(1+z^{2})^{\frac{1}% {4}}}\sum_{k=0}^{\infty}\frac{U_{k}(p)}{\nu^{k}},$ 10.41.4 $\displaystyle K_{\nu}\left(\nu z\right)$ $\displaystyle\sim\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{e^{-\nu\eta}% }{(1+z^{2})^{\frac{1}{4}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{U_{k}(p)}{\nu^{k}},$ 10.41.5 $\displaystyle I_{\nu}'\left(\nu z\right)$ $\displaystyle\sim\frac{(1+z^{2})^{\frac{1}{4}}e^{\nu\eta}}{(2\pi\nu)^{\frac{1}% {2}}z}\sum_{k=0}^{\infty}\frac{V_{k}(p)}{\nu^{k}},$
 10.41.6 $K_{\nu}'\left(\nu z\right)\sim-\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}% \frac{(1+z^{2})^{\frac{1}{4}}e^{-\nu\eta}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{% V_{k}(p)}{\nu^{k}},$

uniformly for $0. Here

 10.41.7 $\eta=(1+z^{2})^{\frac{1}{2}}+\ln\frac{z}{1+(1+z^{2})^{\frac{1}{2}}},$ ⓘ Defines: $\eta$ (locally) Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 9.7.11 Permalink: http://dlmf.nist.gov/10.41.E7 Encodings: TeX, pMML, png See also: Annotations for §10.41(ii), §10.41 and Ch.10
 10.41.8 $p=(1+z^{2})^{-\frac{1}{2}},$ ⓘ Symbols: $z$: complex variable A&S Ref: 9.7.11 Referenced by: §10.41(iii) Permalink: http://dlmf.nist.gov/10.41.E8 Encodings: TeX, pMML, png See also: Annotations for §10.41(ii), §10.41 and Ch.10

where the branches assume their principal values. Also, $U_{k}(p)$ and $V_{k}(p)$ are polynomials in $p$ of degree $3k$, given by $U_{0}(p)=V_{0}(p)=1$, and

 10.41.9 $\displaystyle U_{k+1}(p)$ $\displaystyle=\tfrac{1}{2}p^{2}(1-p^{2})U_{k}^{\prime}(p)+\frac{1}{8}\int_{0}^% {p}(1-5t^{2})U_{k}(t)\,\mathrm{d}t,$ $\displaystyle V_{k+1}(p)$ $\displaystyle=U_{k+1}(p)-\tfrac{1}{2}p(1-p^{2})U_{k}(p)-p^{2}(1-p^{2})U_{k}^{% \prime}(p),$ $k=0,1,2,\dotsc$. ⓘ Defines: $U_{k}(p)$: polynomial coefficient (locally) and $V_{k}(p)$: polynomial coefficient (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $k$: nonnegative integer A&S Ref: 9.3.10, 9.3.14 Permalink: http://dlmf.nist.gov/10.41.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.41(ii), §10.41 and Ch.10

For $k=1,2,3$,

 10.41.10 $\displaystyle U_{1}(p)$ $\displaystyle=\tfrac{1}{24}(3p-5p^{3}),$ $\displaystyle U_{2}(p)$ $\displaystyle=\tfrac{1}{1152}(81p^{2}-462p^{4}+385p^{6}),$ $\displaystyle U_{3}(p)$ $\displaystyle=\tfrac{1}{4\;14720}\*(30375p^{3}-3\;69603p^{5}+7\;65765p^{7}-4\;% 25425p^{9}),$ ⓘ Symbols: $U_{k}(p)$: polynomial coefficient A&S Ref: 9.3.9 Permalink: http://dlmf.nist.gov/10.41.E10 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.41(ii), §10.41 and Ch.10
 10.41.11 $\displaystyle V_{1}(p)$ $\displaystyle=\tfrac{1}{24}(-9p+7p^{3}),$ $\displaystyle V_{2}(p)$ $\displaystyle=\tfrac{1}{1152}(-135p^{2}+594p^{4}-455p^{6}),$ $\displaystyle V_{3}(p)$ $\displaystyle=\tfrac{1}{4\;14720}\*(-42525p^{3}+4\;51737p^{5}-8\;83575p^{7}+4% \;75475p^{9}).$ ⓘ Symbols: $V_{k}(p)$: polynomial coefficient A&S Ref: 9.3.13 Permalink: http://dlmf.nist.gov/10.41.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.41(ii), §10.41 and Ch.10

For $U_{4}(p)$, $U_{5}(p)$, $U_{6}(p)$, see Bickley et al. (1952, p. xxxv).

For numerical tables of $\eta=\eta(z)$ and the coefficients $U_{k}(p)$, $V_{k}(p)$, see Olver (1962, pp. 43–51).

## §10.41(iii) Uniform Expansions for Complex Variable

The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$ $(<\tfrac{1}{2}\pi)$, with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real $z$-axis.

Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the $z$-plane and the $\eta$-plane. The curve $E_{1}BE_{2}$ in the $z$-plane is the upper boundary of the domain $\mathbf{K}$ depicted in Figure 10.20.3 and rotated through an angle $-\tfrac{1}{2}\pi$. Thus $B$ is the point $z=c$, where $c$ is given by (10.20.18).

For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). For extensions of the regions of validity in the $z$-plane and extensions to complex values of $\nu$ see Olver (1997b, pp. 378–382).

For expansions in inverse factorial series see Dunster et al. (1993).

## §10.41(iv) Double Asymptotic Properties

The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. Thus as $z\to\infty$ with $\ell$ $(\geq 1)$ and $\nu$ $(>0)$ both fixed,

 10.41.12 $I_{\nu}\left(\nu z\right)=\frac{e^{\nu\eta}}{(2\pi\nu)^{\frac{1}{2}}(1+z^{2})^% {\frac{1}{4}}}\left(\sum_{k=0}^{\ell-1}\frac{U_{k}(p)}{\nu^{k}}+O\left(\frac{1% }{z^{\ell}}\right)\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$,
 10.41.13 $K_{\nu}\left(\nu z\right)=\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{e^{% -\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\*\left(\sum_{k=0}^{\ell-1}(-1)^{k}\frac{U_% {k}(p)}{\nu^{k}}+O\left(\frac{1}{z^{\ell}}\right)\right),$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.

Similarly for (10.41.5) and (10.41.6).

In the case of (10.41.13) with positive real values of $z$ the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). Then by expanding the quantities $\eta$, $(1+z^{2})^{-\frac{1}{4}}$, and $U_{k}(p)$, $k=0,1,\dotsc,\ell-1$, and rearranging, we arrive at an expansion of the right-hand side of (10.41.13) in powers of $z^{-1}$. Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\ell-1)}$. It also enjoys the same sector of validity.

To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $ze^{\pm\pi i}$ is to replace $\eta$, $(1+z^{2})^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm i(1+z^{2})^{\frac{1}{4}}$, and $-p$, respectively.

## §10.41(v) Double Asymptotic Properties (Continued)

Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. We first prove that for the expansions (10.20.6) for the Hankel functions ${H^{(1)}_{\nu}}\left(\nu z\right)$ and ${H^{(2)}_{\nu}}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to\pm i\infty$, respectively. This is a consequence of the error bounds associated with these expansions. We then extend the validity of this property from $z\to\pm i\infty$ to $z\to\infty$ in the sector $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ in the case of ${H^{(1)}_{\nu}}\left(\nu z\right)$, and to $z\to\infty$ in the sector $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ in the case of ${H^{(2)}_{\nu}}\left(\nu z\right)$. This is done by re-expansion with the aid of (10.20.10), (10.20.11), and §10.41(ii), followed by comparison with (10.17.5) and (10.17.6), with $z$ replaced by $\nu z$. Lastly, we substitute into (10.4.4), again with $z$ replaced by $\nu z$. The final results are:

 10.41.14 $\displaystyle J_{\nu}\left(\nu z\right)$ $\displaystyle=\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\left(\frac{% \operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\left(% \sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+3}% }\right)\right)+\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{% \nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{k}(\zeta)}{\nu^{2k}}+O% \left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),$ 10.41.15 $\displaystyle Y_{\nu}\left(\nu z\right)$ $\displaystyle=-\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\left(\frac{% \operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\left(% \sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+3}% }\right)\right)+\frac{\operatorname{Bi}'\left(\nu^{\frac{2}{3}}\zeta\right)}{% \nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{k}(\zeta)}{\nu^{2k}}+O% \left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),$

as $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta$, or equivalently as $\zeta\to\infty$ in $|\operatorname{ph}\left(-\zeta\right)|\leq\tfrac{2}{3}\pi-\delta$, for fixed $\ell$ $(\geq 0)$ and fixed $\nu$ $(>0)$.

It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when $z\to 0+$ or equivalently $\zeta\to+\infty$. This is because $A_{k}(\zeta)$ and $\zeta^{-\frac{1}{2}}B_{k}(\zeta),k=0,1,\dotsc$, do not form an asymptotic scale (§2.1(v)) as $\zeta\to+\infty$; see Olver (1997b, pp. 422–425).