# §32.11 Asymptotic Approximations for Real Variables

## §32.11(i) First Painlevé Equation

There are solutions of (32.2.1) such that

 32.11.1 $w(x)=-\sqrt{\tfrac{1}{6}|x|}+d|x|^{-1/8}\sin\left(\phi(x)-\theta_{0}\right)+o% \left(|x|^{-1/8}\right),$ $x\to-\infty$,

where

 32.11.2 $\phi(x)=(24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-\tfrac{5}{8}d^{2}\ln|x|\right),$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real, $\phi(z)$: function and $d$: constant Referenced by: §32.11(i) Permalink: http://dlmf.nist.gov/32.11.E2 Encodings: TeX, pMML, png See also: Annotations for §32.11(i), §32.11 and Ch.32

and $d$ and $\theta_{0}$ are constants.

There are also solutions of (32.2.1) such that

 32.11.3 $w(x)\sim\sqrt{\tfrac{1}{6}|x|},$ $x\to-\infty$. ⓘ Symbols: $\sim$: asymptotic equality and $x$: real Permalink: http://dlmf.nist.gov/32.11.E3 Encodings: TeX, pMML, png See also: Annotations for §32.11(i), §32.11 and Ch.32

Next, for given initial conditions $w(0)=0$ and $w^{\prime}(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis. There are two special values of $k$, $k_{1}$ and $k_{2}$, with the properties $-0.45142\;8, $1.85185\;3, and such that:

1. (a)

If $k, then $w(x)>0$ for $x_{0}, where $x_{0}$ is the first pole on the negative real axis.

2. (b)

If $k_{1}, then $w(x)$ oscillates about, and is asymptotic to, $-\sqrt{\tfrac{1}{6}|x|}$ as $x\to-\infty$.

3. (c)

If $k_{2}, then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_{0}$ to $0$.

For illustration see Figures 32.3.1 to 32.3.4, and for further information see Joshi and Kitaev (2005), Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and Kitaev (1994).

## §32.11(ii) Second Painlevé Equation

Consider the special case of $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$:

 32.11.4 $w^{\prime\prime}=2w^{3}+xw,$ ⓘ Symbols: $x$: real Referenced by: §32.11(ii), §32.11(iii) Permalink: http://dlmf.nist.gov/32.11.E4 Encodings: TeX, pMML, png See also: Annotations for §32.11(ii), §32.11 and Ch.32

with boundary condition

 32.11.5 $w(x)\to 0,$ $x\to+\infty$. ⓘ Symbols: $x$: real Referenced by: §32.11(ii) Permalink: http://dlmf.nist.gov/32.11.E5 Encodings: TeX, pMML, png See also: Annotations for §32.11(ii), §32.11 and Ch.32

Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). Conversely, for any nonzero real $k$, there is a unique solution $w_{k}(x)$ of (32.11.4) that is asymptotic to $k\mathrm{Ai}\left(x\right)$ as $x\to+\infty$.

If $|k|<1$, then $w_{k}(x)$ exists for all sufficiently large $|x|$ as $x\to-\infty$, and

 32.11.6 $w_{k}(x)=d|x|^{-1/4}\sin\left(\phi(x)-\theta_{0}\right)+o\left(|x|^{-1/4}% \right),$

where

 32.11.7 $\phi(x)=\tfrac{2}{3}|x|^{3/2}-\tfrac{3}{4}d^{2}\ln|x|,$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real, $\phi(z)$: function and $d$: constant Permalink: http://dlmf.nist.gov/32.11.E7 Encodings: TeX, pMML, png See also: Annotations for §32.11(ii), §32.11 and Ch.32

and $d$ $(\neq 0)$, $\theta_{0}$ are real constants. Connection formulas for $d$ and $\theta_{0}$ are given by

 32.11.8 $d^{2}=-\pi^{-1}\ln\left(1-k^{2}\right),$
 32.11.9 $\theta_{0}=\tfrac{3}{2}d^{2}\ln 2+\operatorname{ph}\Gamma\left(1-\tfrac{1}{2}% id^{2}\right)+\tfrac{1}{4}\pi(1-2\operatorname{sign}\left(k\right)),$

where $\Gamma$ is the gamma function (§5.2(i)), and the branch of the $\operatorname{ph}$ function is immaterial.

If $|k|=1$, then

 32.11.10 $w_{k}(x)\sim\operatorname{sign}\left(k\right)\sqrt{\tfrac{1}{2}|x|},$ $x\to-\infty$. ⓘ Symbols: $\sim$: asymptotic equality, $\operatorname{sign}\NVar{x}$: sign of, $x$: real and $k$: real Permalink: http://dlmf.nist.gov/32.11.E10 Encodings: TeX, pMML, png See also: Annotations for §32.11(ii), §32.11 and Ch.32

If $|k|>1$, then $w_{k}(x)$ has a pole at a finite point $x=c_{0}$, dependent on $k$, and

 32.11.11 $w_{k}(x)\sim\operatorname{sign}\left(k\right)(x-c_{0})^{-1},$ $x\to c_{0}+$. ⓘ Symbols: $\sim$: asymptotic equality, $\operatorname{sign}\NVar{x}$: sign of, $x$: real, $k$: real and $c_{0}$: pole Permalink: http://dlmf.nist.gov/32.11.E11 Encodings: TeX, pMML, png See also: Annotations for §32.11(ii), §32.11 and Ch.32

For illustration see Figures 32.3.5 and 32.3.6, and for further information see Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988), Deift and Zhou (1995), Segur and Ablowitz (1981), and Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

## §32.11(iii) Modified Second Painlevé Equation

Replacement of $w$ by $iw$ in (32.11.4) gives

 32.11.12 $w^{\prime\prime}=-2w^{3}+xw.$ ⓘ Symbols: $x$: real Referenced by: §32.11(iii) Permalink: http://dlmf.nist.gov/32.11.E12 Encodings: TeX, pMML, png See also: Annotations for §32.11(iii), §32.11 and Ch.32

Any nontrivial real solution of (32.11.12) satisfies

 32.11.13 $w(x)=d|x|^{-1/4}\sin\left(\phi(x)-\chi\right)+O\left(|x|^{-5/4}\ln|x|\right),$ $x\to-\infty$,

where

 32.11.14 $\phi(x)=\tfrac{2}{3}|x|^{3/2}+\tfrac{3}{4}d^{2}\ln|x|,$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real, $\phi(z)$: function and $d$: constant Permalink: http://dlmf.nist.gov/32.11.E14 Encodings: TeX, pMML, png See also: Annotations for §32.11(iii), §32.11 and Ch.32

with $d$ $(\neq 0)$ and $\chi$ arbitrary real constants.

In the case when

 32.11.15 $\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-\operatorname{ph}\Gamma\left(% \tfrac{1}{2}id^{2}\right)=n\pi,$

with $n\in\mathbb{Z}$, we have

 32.11.16 $w(x)\sim k\mathrm{Ai}\left(x\right),$ $x\to+\infty$, ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\sim$: asymptotic equality, $x$: real and $k$: real Permalink: http://dlmf.nist.gov/32.11.E16 Encodings: TeX, pMML, png See also: Annotations for §32.11(iii), §32.11 and Ch.32

where $k$ is a nonzero real constant. The connection formulas for $k$ are

 32.11.17 $d^{2}=\pi^{-1}\ln\left(1+k^{2}\right),$ $\operatorname{sign}\left(k\right)=(-1)^{n}$.

In the generic case

 32.11.18 $\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-\operatorname{ph}\Gamma\left(% \tfrac{1}{2}id^{2}\right)\neq n\pi,$

we have

 32.11.19 $w(x)=\sigma\sqrt{\tfrac{1}{2}x}+\sigma\rho(2x)^{-1/4}\cos\left(\psi(x)+\theta% \right)+O\left(x^{-1}\right),$ $x\to+\infty$,

where $\sigma$, $\rho$ $(>0)$, and $\theta$ are real constants, and

 32.11.20 $\psi(x)=\tfrac{2}{3}\sqrt{2}x^{3/2}-\tfrac{3}{2}\rho^{2}\ln x.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real, $\rho$: real constant and $\psi(z)$: function Permalink: http://dlmf.nist.gov/32.11.E20 Encodings: TeX, pMML, png See also: Annotations for §32.11(iii), §32.11 and Ch.32

The connection formulas for $\sigma$, $\rho$, and $\theta$ are

 32.11.21 $\sigma=-\operatorname{sign}\left(\Im s\right),$ ⓘ Symbols: $\Im$: imaginary part, $\operatorname{sign}\NVar{x}$: sign of, $\sigma$: real constant and $s$ Permalink: http://dlmf.nist.gov/32.11.E21 Encodings: TeX, pMML, png See also: Annotations for §32.11(iii), §32.11 and Ch.32
 32.11.22 $\rho^{2}=\pi^{-1}\ln\left((1+|s|^{2})/|2\Im s|\right),$
 32.11.23 $\theta=-\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}+\operatorname{ph}\left(1+s^% {2}\right)+\operatorname{ph}\Gamma\left(i\rho^{2}\right),$

where

 32.11.24 $s=\left(\exp\left(\pi d^{2}\right)-1\right)^{1/2}\*\exp\left(i\left(\tfrac{3}{% 2}d^{2}\ln 2-\tfrac{1}{4}\pi+\chi-\operatorname{ph}\Gamma\left(\tfrac{1}{2}id^% {2}\right)\right)\right).$

## §32.11(iv) Third Painlevé Equation

For $\mbox{P}_{\mbox{\scriptsize III}}$, with $\alpha=-\beta=2\nu$ $(\in\mathbb{R})$ and $\gamma=-\delta=1$,

 32.11.25 $w(x)-1\sim-\lambda\Gamma\left(\nu+\tfrac{1}{2}\right)2^{-2\nu}x^{-\nu-(1/2)}e^% {-2x},$ $x\to+\infty$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $x$: real and $\nu$: parameter Referenced by: §32.11(iv) Permalink: http://dlmf.nist.gov/32.11.E25 Encodings: TeX, pMML, png See also: Annotations for §32.11(iv), §32.11 and Ch.32

where $\lambda$ is an arbitrary constant such that $-1/\pi<\lambda<1/\pi$, and

 32.11.26 $w(x)\sim Bx^{\sigma},$ $x\to 0$, ⓘ Symbols: $\sim$: asymptotic equality, $x$: real, $B$: constant and $\sigma$: constant Referenced by: §32.11(iv) Permalink: http://dlmf.nist.gov/32.11.E26 Encodings: TeX, pMML, png See also: Annotations for §32.11(iv), §32.11 and Ch.32

where $B$ and $\sigma$ are arbitrary constants such that $B\neq 0$ and $|\Re\sigma|<1$. The connection formulas relating (32.11.25) and (32.11.26) are

 32.11.27 $\sigma=(2/\pi)\operatorname{arcsin}\left(\pi\lambda\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arcsin}\NVar{z}$: arcsine function and $\sigma$: constant Permalink: http://dlmf.nist.gov/32.11.E27 Encodings: TeX, pMML, png See also: Annotations for §32.11(iv), §32.11 and Ch.32
 32.11.28 $B=2^{-2\sigma}\frac{{\Gamma}^{2}\left(\tfrac{1}{2}(1-\sigma)\right)\Gamma\left% (\tfrac{1}{2}(1+\sigma)+\nu\right)}{{\Gamma}^{2}\left(\tfrac{1}{2}(1+\sigma)% \right)\Gamma\left(\tfrac{1}{2}(1-\sigma)+\nu\right)}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\nu$: parameter, $B$: constant and $\sigma$: constant Permalink: http://dlmf.nist.gov/32.11.E28 Encodings: TeX, pMML, png See also: Annotations for §32.11(iv), §32.11 and Ch.32

See also Abdullaev (1985), Novokshënov (1985), Its and Novokshënov (1986), Kitaev (1987), Bobenko (1991), Bobenko and Its (1995), Tracy and Widom (1997), and Kitaev and Vartanian (2004).

## §32.11(v) Fourth Painlevé Equation

Consider $\mbox{P}_{\mbox{\scriptsize IV}}$ with $\alpha=2\nu+1$ $(\in\mathbb{R})$ and $\beta=0$, that is,

 32.11.29 $w^{\prime\prime}=\frac{(w^{\prime})^{2}}{2w}+\frac{3}{2}w^{3}+4xw^{2}+2(x^{2}-% 2\nu-1)w,$ ⓘ Symbols: $x$: real and $\nu$: parameter Referenced by: §32.11(v) Permalink: http://dlmf.nist.gov/32.11.E29 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

and with boundary condition

 32.11.30 $w(x)\to 0,$ $x\to+\infty$. ⓘ Symbols: $x$: real Referenced by: §32.11(v) Permalink: http://dlmf.nist.gov/32.11.E30 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{U}^{2}\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$, where $h$ $(\neq 0)$ is a constant. Conversely, for any $h$ $(\neq 0)$ there is a unique solution $w_{h}(x)$ of (32.11.29) that is asymptotic to $h{U}^{2}\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$. Here $U$ denotes the parabolic cylinder function (§12.2).

Now suppose $x\to-\infty$. If $0\leq h, where

 32.11.31 $h^{*}=\ifrac{1}{\left(\pi^{1/2}\Gamma\left(\nu+1\right)\right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\nu$: parameter Permalink: http://dlmf.nist.gov/32.11.E31 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

then $w_{h}(x)$ has no poles on the real axis. Furthermore, if $\nu=n=0,1,2,\dots$, then

 32.11.32 $w_{h}(x)\sim h2^{n}x^{2n}\exp\left(-x^{2}\right),$ $x\to-\infty$. ⓘ Symbols: $\sim$: asymptotic equality, $\exp\NVar{z}$: exponential function, $n$: integer and $x$: real Permalink: http://dlmf.nist.gov/32.11.E32 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

Alternatively, if $\nu$ is not zero or a positive integer, then

 32.11.33 $w_{h}(x)=-\tfrac{2}{3}x+\tfrac{4}{3}d\sqrt{3}\sin\left(\phi(x)-\theta_{0}% \right)+O\left(x^{-1}\right),$ $x\to-\infty$,

where

 32.11.34 $\phi(x)=\tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\ln\left(\sqrt{2}|x% |\right),$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $x$: real, $\phi(z)$: function and $d$: constant Permalink: http://dlmf.nist.gov/32.11.E34 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

and $d$ $(>0)$ and $\theta_{0}$ are real constants. Connection formulas for $d$ and $\theta_{0}$ are given by

 32.11.35 $\displaystyle d^{2}$ $\displaystyle=-\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln\left(1-|\mu|^{2}\right),$ ⓘ 32.11.36 $\displaystyle\theta_{0}$ $\displaystyle=\tfrac{1}{3}d^{2}\sqrt{3}\ln 3+\tfrac{2}{3}\pi\nu+\tfrac{7}{12}% \pi+\operatorname{ph}\mu+\operatorname{ph}\Gamma\left(-\tfrac{2}{3}i\sqrt{3}d^% {2}\right),$

where

 32.11.37 $\mu=1+\left(\ifrac{2ih\pi^{3/2}\exp\left(-i\pi\nu\right)}{\Gamma\left(-\nu% \right)}\right),$

and the branch of the $\operatorname{ph}$ function is immaterial.

Next if $h=h^{*}$, then

 32.11.38 $w_{h^{*}}(x)\sim-2x,$ $x\to-\infty$, ⓘ Symbols: $\sim$: asymptotic equality and $x$: real Permalink: http://dlmf.nist.gov/32.11.E38 Encodings: TeX, pMML, png See also: Annotations for §32.11(v), §32.11 and Ch.32

and $w_{h^{*}}(x)$ has no poles on the real axis.

Lastly if $h>h^{*}$, then $w_{h}(x)$ has a simple pole on the real axis, whose location is dependent on $h$.

For illustration see Figures 32.3.732.3.10. In terms of the parameter $k$ that is used in these figures $h=2^{3/2}k^{2}$.