# §10.20 Uniform Asymptotic Expansions for Large Order

## §10.20(i) Real Variables

Define $\zeta=\zeta(z)$ to be the solution of the differential equation

 10.20.1 $\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{2}=\frac{1-z^{2}}{\zeta z^{2}}$

that is infinitely differentiable on the interval $0, including $z=1$. Then

 10.20.2 $\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\,\mathrm{d% }t=\ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2}},$ $0, ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $z$: complex variable and $\zeta(z)$: solution A&S Ref: 9.3.38 Referenced by: §10.20(ii) Permalink: http://dlmf.nist.gov/10.20.E2 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10
 10.20.3 $\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\,% \mathrm{d}t=\sqrt{z^{2}-1}-\operatorname{arcsec}z,$ $1\leq z<\infty$, ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\operatorname{arcsec}\NVar{z}$: arcsecant function, $z$: complex variable and $\zeta(z)$: solution A&S Ref: 9.3.39 Referenced by: §10.20(ii), §10.21(viii) Permalink: http://dlmf.nist.gov/10.20.E3 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

all functions taking their principal values, with $\zeta=\infty,0,-\infty$, corresponding to $z=0,1,\infty$, respectively.

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

 10.20.4 $\displaystyle J_{\nu}\left(\nu z\right)$ $\displaystyle\sim\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\*\left(% \frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}% \sum_{k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{Ai}'\left% (\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B_{% k}(\zeta)}{\nu^{2k}}\right),$ 10.20.5 $\displaystyle Y_{\nu}\left(\nu z\right)$ $\displaystyle\sim-\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\left(\frac% {\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_% {k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{Bi}'\left(\nu^% {\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B_{k}(% \zeta)}{\nu^{2k}}\right),$
 10.20.6 $\rselection{{H^{(1)}_{\nu}}\left(\nu z\right)\\ {H^{(2)}_{\nu}}\left(\nu z\right)}\sim 2e^{\mp\pi i/3}\left(\frac{4\zeta}{1-z^% {2}}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(e^{\pm 2\pi i/3}% \nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k% }(\zeta)}{\nu^{2k}}+\frac{e^{\pm 2\pi i/3}\operatorname{Ai}'\left(e^{\pm 2\pi i% /3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B% _{k}(\zeta)}{\nu^{2k}}\right),$
 10.20.7 $\displaystyle J_{\nu}'\left(\nu z\right)$ $\displaystyle\sim-\frac{2}{z}\left(\frac{1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}% \*\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac% {4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{% Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{2}{3}}}\sum_{k=0}^{\infty}% \frac{D_{k}(\zeta)}{\nu^{2k}}\right),$ 10.20.8 $\displaystyle Y_{\nu}'\left(\nu z\right)$ $\displaystyle\sim\frac{2}{z}\left(\frac{1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}% \*\left(\frac{\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac% {4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{% Bi}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{2}{3}}}\sum_{k=0}^{\infty}% \frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
 10.20.9 $\rselection{{H^{(1)}_{\nu}}'\left(\nu z\right)\\ {H^{(2)}_{\nu}}'\left(\nu z\right)}\sim\frac{4e^{\mp 2\pi i/3}}{z}\left(\frac{% 1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}\*\left(\frac{e^{\mp 2\pi i/3}% \operatorname{Ai}\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{% \operatorname{Ai}'\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{2}{3}}}\sum_{k=0}^{\infty}\frac{D_{k}(\zeta)}{\nu^{2k}}\right),$

uniformly for $z$ $\in(0,\infty)$ in all cases, where $\operatorname{Ai}$ and $\operatorname{Bi}$ are the Airy functions (§9.2).

In the following formulas for the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, $u_{k}$, $v_{k}$ are the constants defined in §9.7(i), and $U_{k}(p)$, $V_{k}(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii).

### Interval $0

 10.20.10 $A_{k}(\zeta)=\sum_{j=0}^{2k}(\tfrac{3}{2})^{j}v_{j}\zeta^{-3j/2}U_{2k-j}\left(% (1-z^{2})^{-\frac{1}{2}}\right),$ ⓘ Defines: $A_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $v_{k}$: constants and $U_{k}(p)$: polynomial coefficient A&S Ref: 9.3.40 Referenced by: §10.20(i), §10.41(v), §10.74(i) Permalink: http://dlmf.nist.gov/10.20.E10 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10
 10.20.11 $B_{k}(\zeta)=-\zeta^{-\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}u_{j}% \zeta^{-3j/2}U_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$ ⓘ Defines: $B_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $u_{k}$: constants and $U_{k}(p)$: polynomial coefficient A&S Ref: 9.3.40 Referenced by: §10.21(viii), §10.41(v) Permalink: http://dlmf.nist.gov/10.20.E11 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10
 10.20.12 $C_{k}(\zeta)=-\zeta^{\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}v_{j}\zeta% ^{-3j/2}V_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$ ⓘ Defines: $C_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $v_{k}$: constants and $V_{k}(p)$: polynomial coefficient A&S Ref: 9.3.46 Referenced by: §10.21(viii) Permalink: http://dlmf.nist.gov/10.20.E12 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10
 10.20.13 $D_{k}(\zeta)=\sum_{j=0}^{2k}(\tfrac{3}{2})^{j}u_{j}\zeta^{-3j/2}V_{2k-j}\left(% (1-z^{2})^{-\frac{1}{2}}\right).$ ⓘ Defines: $D_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $u_{k}$: constants and $V_{k}(p)$: polynomial coefficient A&S Ref: 9.3.46 Referenced by: §10.20(i), §10.74(i) Permalink: http://dlmf.nist.gov/10.20.E13 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

### Interval $1

In formulas (10.20.10)–(10.20.13) replace $\zeta^{\frac{1}{2}}$, $\zeta^{-\frac{1}{2}}$, $\zeta^{-3j/2}$, and $(1-z^{2})^{-\frac{1}{2}}$ by $-i(-\zeta)^{\frac{1}{2}}$, $i(-\zeta)^{-\frac{1}{2}}$, $i^{3j}(-\zeta)^{-3j/2}$, and $i(z^{2}-1)^{-\frac{1}{2}}$, respectively.

Note: Another way of arranging the above formulas for the coefficients $A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)$, and $D_{k}(\zeta)$ would be by analogy with (12.10.42) and (12.10.46). In this way there is less usage of many-valued functions.

### Values at $\zeta=0$

 10.20.14 $\displaystyle A_{0}(0)$ $\displaystyle=1,$ $\displaystyle A_{1}(0)$ $\displaystyle=-\tfrac{1}{225},$ $\displaystyle A_{2}(0)$ $\displaystyle=\tfrac{1\;51439}{2182\;95000},$ $\displaystyle A_{3}(0)$ $\displaystyle=-\tfrac{8872\;78009}{250\;49351\;25000},\\$ $\displaystyle B_{0}(0)$ $\displaystyle=\tfrac{1}{70}2^{\frac{1}{3}},$ $\displaystyle B_{1}(0)$ $\displaystyle=-\tfrac{1213}{10\;23750}2^{\frac{1}{3}},$ $\displaystyle B_{2}(0)$ $\displaystyle=\tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}},$ $\displaystyle B_{3}(0)$ $\displaystyle=-\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3% }}.$ ⓘ Symbols: $A_{k}(\zeta)$: coefficients and $B_{k}(\zeta)$: coefficients Referenced by: §10.20(i), Erratum (V1.0.5) for Equation (10.20.14) Permalink: http://dlmf.nist.gov/10.20.E14 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png Errata (effective with 1.0.5): Originally the value given for $B_{3}(0)$ was given incorrectly as $B_{3}(0)=-\tfrac{430\;99056\;39368\;59253}{5\;68167\;34399\;42500\;00000}2^{% \frac{1}{3}}$. Reported 2012-05-11 by Antony Lee See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

Each of the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, $k=0,1,2,\dotsc$, is real and infinitely differentiable on the interval $-\infty<\zeta<\infty$. For (10.20.14) and further information on the coefficients see Temme (1997).

For numerical tables of $\zeta=\zeta(z)$, $(4\zeta/(1-z^{2}))^{\frac{1}{4}}$ and $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$ see Olver (1962, pp. 28–42).

## §10.20(ii) Complex Variables

The function $\zeta=\zeta(z)$ given by (10.20.2) and (10.20.3) can be continued analytically to the $z$-plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.

The equations of the curved boundaries $D_{1}E_{1}$ and $D_{2}E_{2}$ in the $\zeta$-plane are given parametrically by

 10.20.15 $\zeta=(\tfrac{3}{2})^{\frac{2}{3}}(\tau\mp i\pi)^{\frac{2}{3}},$ $0\leq\tau<\infty$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\zeta(z)$: solution Permalink: http://dlmf.nist.gov/10.20.E15 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

respectively.

The curves $BP_{1}E_{1}$ and $BP_{2}E_{2}$ in the $z$-plane are the inverse maps of the line segments

 10.20.16 $\zeta=e^{\mp i\pi/3}\tau,$ $0\leq\tau\leq(\tfrac{3}{2}\pi)^{\frac{2}{3}}$,

respectively. They are given parametrically by

 10.20.17 $z=\pm(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}\pm\mathrm{i}(\tau^{2}-\tau\tanh% \tau)^{\frac{1}{2}},$ $0\leq\tau\leq\tau_{0}$,

where $\tau_{0}=1.19968\ldots$ is the positive root of the equation $\tau=\coth\tau$. The points $P_{1},P_{2}$ where these curves intersect the imaginary axis are $\pm ic$, where

 10.20.18 $c=(\tau_{0}^{2}-1)^{\frac{1}{2}}=0.66274\dotsc.$ ⓘ Defines: $c$ (locally) Referenced by: §10.41(iii) Permalink: http://dlmf.nist.gov/10.20.E18 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

The eye-shaped closed domain in the uncut $z$-plane that is bounded by $BP_{1}E_{1}$ and $BP_{2}E_{2}$ is denoted by $\mathbf{K}$; see Figure 10.20.3.

As $\nu\to\infty$ through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for $|\operatorname{ph}z|\leq\pi-\delta$, the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, being the analytic continuations of the functions defined in §10.20(i) when $\zeta$ is real.

For proofs of the above results and for error bounds and extensions of the regions of validity see Olver (1997b, pp. 419–425). For extensions to complex $\nu$ see Olver (1954). For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004).

## §10.20(iii) Double Asymptotic Properties

For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).