§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 $\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\mathrm{d}t% =\mathop{\ln\/}\nolimits\!\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2% }},$ $0, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 10.20.3 $\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\mathrm{% d}t=\sqrt{z^{2}-1}-\mathop{\mathrm{arcsec}\/}\nolimits z,$ $1\leq z<\infty$, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arcsec}\/}\nolimits\NVar{z}$: 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)

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

uniformly for $z$ $\in(0,\infty)$ in all cases, where $\mathop{\mathrm{Ai}\/}\nolimits$ and $\mathop{\mathrm{Bi}\/}\nolimits$ 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 $U_{k}(p)$: polynomials 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)
 10.20.11 $B_{k}(\zeta)=-\zeta^{-\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}u_{j}% \zeta^{-3j/2}U_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$ Defines: $B_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $u_{k}$: constants and $U_{k}(p)$: polynomials 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)
 10.20.12 $C_{k}(\zeta)=-\zeta^{\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}v_{j}\zeta% ^{-3j/2}V_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$ Defines: $C_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $v_{k}$: constants and $V_{k}(p)$: polynomials 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)
 10.20.13 $D_{k}(\zeta)=\sum_{j=0}^{2k}(\tfrac{3}{2})^{j}u_{j}\zeta^{-3j/2}V_{2k-j}\left(% (1-z^{2})^{-\frac{1}{2}}\right).$ Defines: $D_{k}(\zeta)$: coefficients (locally) Symbols: $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: solution, $u_{k}$: constants and $V_{k}(p)$: polynomials 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)

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), 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)

Each of the coefficients $A_{k}(\zeta)$, $B_{k}(\zeta)$, $C_{k}(\zeta)$, and $D_{k}(\zeta)$, $k=0,1,2,\ldots$, 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 and $\zeta(z)$: solution Permalink: http://dlmf.nist.gov/10.20.E15 Encodings: TeX, pMML, png See also: Annotations for 10.20(ii)

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 $\zeta=e^{\mp i\pi/3}\tau,$ $0\leq\tau\leq(\tfrac{3}{2}\pi)^{\frac{2}{3}}$, Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $\zeta(z)$: solution Permalink: http://dlmf.nist.gov/10.20.E16 Encodings: TeX, pMML, png See also: Annotations for 10.20(ii)

respectively. They are given parametrically by

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

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

 10.20.18 $c=(\tau_{0}^{2}-1)^{\frac{1}{2}}=0.66274\ldots.$ 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)

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 $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$, 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).