32 Painlevé Transcendents Properties 32.9 Other Elementary Solutions 32.11 Asymptotic Approximations for Real Variables

§32.10 Special Function Solutions

Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.10

§32.10(i) Introduction
§32.10(ii) Second Painlevé Equation
§32.10(iii) Third Painlevé Equation
§32.10(iv) Fourth Painlevé Equation
§32.10(v) Fifth Painlevé Equation
§32.10(vi) Sixth Painlevé Equation

§32.10(i) Introduction

Notes:: See Gromak (1978); also Albrecht et al. (1996), Gromak et al. (2002).
Permalink:: http://dlmf.nist.gov/32.10.i

For certain combinations of the parameters, $\mbox{P}_{{\mbox{\scriptsize II}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. All solutions of $\mbox{P}_{{\mbox{\scriptsize II}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ that are expressible in terms of special functions satisfy a first-order equation of the form

32.10.1 $(w^{{\prime}})^{n}+\sum_{{j=0}}^{{n-1}}F_{j}(w,z)(w^{{\prime}})^{j}=0,$

Symbols:: : integer and : real
Permalink:: http://dlmf.nist.gov/32.10.E1
Encodings:: TeX, pMML, png

where $F_{j}(w,z)$ is polynomial in with coefficients that are rational functions of .

§32.10(ii) Second Painlevé Equation

Notes:: See Gambier (1910); also Airault (1979), Gromak (1978), Okamoto (1986), and Flaschka and Newell (1980).
Keywords:: Airy functions, Painlevé equations
Permalink:: http://dlmf.nist.gov/32.10.ii

$\mbox{P}_{{\mbox{\scriptsize II}}}$ has solutions expressible in terms of Airy functions (§9.2) iff

32.10.2 $\alpha=n+\tfrac{1}{2},$

Symbols:: : integer and $\alpha$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.10.E2
Encodings:: TeX, pMML, png

with $n\in\Integer$ . For example, if $\alpha=\tfrac{1}{2}\varepsilon$ , with $\varepsilon=\pm 1$ , then the Riccati equation is

32.10.3 $\varepsilon w^{{\prime}}=w^{2}+\tfrac{1}{2}z,$

Symbols:: : real and $\varepsilon=\pm 1$
Permalink:: http://dlmf.nist.gov/32.10.E3
Encodings:: TeX, pMML, png

with solution

32.10.4 $w(z;\tfrac{1}{2}\varepsilon)=-\varepsilon\phi^{{\prime}}(z)/\phi(z),$

Symbols:: : real, $\varepsilon=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E4
Encodings:: TeX, pMML, png

where

32.10.5 $\phi(z)=C_{1}\mathop{\mathrm{Ai}\/}\nolimits\!\left(-2^{{-1/3}}z\right)+C_{2}% \mathop{\mathrm{Bi}\/}\nolimits\!\left(-2^{{-1/3}}z\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : real, $\phi(z)$ : function and $C_{j}$ : constants
Referenced by:: §32.10(ii)
Permalink:: http://dlmf.nist.gov/32.10.E5
Encodings:: TeX, pMML, png

with $C_{1}$ , $C_{2}$ arbitrary constants.

Solutions for other values of $\alpha$ are derived from $w(z;\pm\tfrac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). For example,

32.10.6 $w(z;\tfrac{3}{2})=\Phi-\dfrac{1}{2\Phi^{2}+z},$

Symbols:: : real and $\Phi$
Permalink:: http://dlmf.nist.gov/32.10.E6
Encodings:: TeX, pMML, png

32.10.7 $w(z;\tfrac{5}{2})=\dfrac{1}{2\Phi^{2}+z}+\dfrac{2z\Phi^{2}+\Phi+z^{2}}{4\Phi^{% 3}+2z\Phi-1},$

Symbols:: : real and $\Phi$
Permalink:: http://dlmf.nist.gov/32.10.E7
Encodings:: TeX, pMML, png

where $\Phi=\phi^{{\prime}}(z)/\phi(z)$ , with $\phi(z)$ given by (32.10.5).

More generally, if $n=1,2,3,\dots$ , then

32.10.8 $w(z;n+\tfrac{1}{2})=\frac{d}{dz}\left(\mathop{\ln\/}\nolimits\!\left(\frac{% \tau_{{n}}(z)}{\tau_{{n+1}}(z)}\right)\right),$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : integer, : real and $\tau_{n}(z)$ : determinant
Permalink:: http://dlmf.nist.gov/32.10.E8
Encodings:: TeX, pMML, png

where $\tau_{n}(z)$ is the $n\times n$ determinant

32.10.9 $\tau_{n}(z)=\begin{vmatrix}\phi(z)&\phi^{{\prime}}(z)&\cdots&\phi^{{(n-1)}}(z)% \\ \phi^{{\prime}}(z)&\phi^{{\prime\prime}}(z)&\cdots&\phi^{{(n)}}(z)\\ \vdots&\vdots&\ddots&\vdots\\ \phi^{{(n-1)}}(z)&\phi^{{(n)}}(z)&\cdots&\phi^{{(2n-2)}}(z)\end{vmatrix},$

Symbols:: : integer, : real, $\phi(z)$ : function and $\tau_{n}(z)$ : determinant
Permalink:: http://dlmf.nist.gov/32.10.E9
Encodings:: TeX, pMML, png

and

32.10.10 $w(z;-n-\tfrac{1}{2})=-w(z;n+\tfrac{1}{2}).$

Symbols:: : integer and : real
Permalink:: http://dlmf.nist.gov/32.10.E10
Encodings:: TeX, pMML, png

§32.10(iii) Third Painlevé Equation

Notes:: See Gromak (1978), Lukaševič (1965, 1967b), Mansfield and Webster (1998), Umemura and Watanabe (1998); also Gromak et al. (2002, §35).
Keywords:: Bessel functions, Painlevé equations
Permalink:: http://dlmf.nist.gov/32.10.iii

If $\gamma\delta\neq 0$ , then as in §32.2(ii) we may set $\gamma=1$ and $\delta=-1$ . $\mbox{P}_{{\mbox{\scriptsize III}}}$ then has solutions expressible in terms of Bessel functions (§10.2) iff

32.10.11 $\varepsilon_{1}\alpha+\varepsilon_{2}\beta=4n+2,$

Symbols:: : integer, $\varepsilon_{j}=\pm 1$ , $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.10.E11
Encodings:: TeX, pMML, png

with $n\in\Integer$ , and $\varepsilon_{1}=\pm 1$ , $\varepsilon_{2}=\pm 1$ , independently. In the case $\varepsilon_{1}\alpha+\varepsilon_{2}\beta=2$ , the Riccati equation is

32.10.12 $zw^{{\prime}}=\varepsilon_{1}zw^{2}+(\alpha\varepsilon_{1}-1)w+\varepsilon_{2}z.$

Symbols:: : real, $\varepsilon_{j}=\pm 1$ and $\alpha$ : arbitrary constant
Referenced by:: §32.10(iii)
Permalink:: http://dlmf.nist.gov/32.10.E12
Encodings:: TeX, pMML, png

If $\alpha\neq\varepsilon_{1}$ , then (32.10.12) has the solution

32.10.13 $w(z)=-\varepsilon_{1}\phi^{{\prime}}(z)/\phi(z),$

Symbols:: : real, $\varepsilon_{j}=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E13
Encodings:: TeX, pMML, png

where

32.10.14 $\phi(z)=z^{\nu}\left(C_{1}\mathop{J_{{\nu}}\/}\nolimits\!\left(\zeta\right)+C_% {2}\mathop{Y_{{\nu}}\/}\nolimits\!\left(\zeta\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, : real, $\phi(z)$ : function, $\nu$ and $C_{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.10.E14
Encodings:: TeX, pMML, png

with $\zeta=\sqrt{\varepsilon_{1}\varepsilon_{2}}z$ , $\nu=\tfrac{1}{2}\alpha\varepsilon_{1}$ , and $C_{1}$ , $C_{2}$ arbitrary constants.

For examples and plots see Milne et al. (1997). For determinantal representations see Forrester and Witte (2002) and Okamoto (1987c).

§32.10(iv) Fourth Painlevé Equation

Notes:: See Gambier (1910), Gromak (1987), Gromak and Lukaševič (1982), Lukaševič (1965, 1967a); also Gromak et al. (2002, Chapter 6).
Keywords:: Hermite polynomials, Painlevé equations, parabolic cylinder functions
Referenced by:: §32.8(iv)
Permalink:: http://dlmf.nist.gov/32.10.iv

$\mbox{P}_{{\mbox{\scriptsize IV}}}$ has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either

32.10.15 $\beta=-2(2n+1+\varepsilon\alpha)^{2},$

Symbols:: : integer, $\varepsilon=\pm 1$ , $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Referenced by:: §32.10(iv)
Permalink:: http://dlmf.nist.gov/32.10.E15
Encodings:: TeX, pMML, png

32.10.16 $\beta=-2n^{2},$

Symbols:: : integer and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.10.E16
Encodings:: TeX, pMML, png

with $n\in\Integer$ and $\varepsilon=\pm 1$ . In the case when in (32.10.15), the Riccati equation is

32.10.17 $w^{{\prime}}=\varepsilon(w^{2}+2zw)-2(1+\varepsilon\alpha),$

Symbols:: : real, $\varepsilon=\pm 1$ and $\alpha$ : arbitrary constant
Referenced by:: §32.10(iv)
Permalink:: http://dlmf.nist.gov/32.10.E17
Encodings:: TeX, pMML, png

which has the solution

32.10.18 $w(z)=-\varepsilon\phi^{{\prime}}(z)/\phi(z),$

Symbols:: : real, $\varepsilon=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E18
Encodings:: TeX, pMML, png

where

32.10.19 $\phi(z)=\left(C_{1}\mathop{U\/}\nolimits\!\left(a,\sqrt{2}z\right)+C_{2}% \mathop{V\/}\nolimits\!\left(a,\sqrt{2}z\right)\right)\mathop{\exp\/}\nolimits% \!\left(\tfrac{1}{2}\varepsilon z^{2}\right),$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real, $\varepsilon=\pm 1$ , $\phi(z)$ : function, and : constant
Permalink:: http://dlmf.nist.gov/32.10.E19
Encodings:: TeX, pMML, png

with $a=\alpha+\tfrac{1}{2}\varepsilon$ , and $C_{1}$ , $C_{2}$ arbitrary constants. When $a+\tfrac{1}{2}$ is zero or a negative integer the $\mathop{U\/}\nolimits$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus

32.10.20 $w(z;-m,-2(m-1)^{2})=-\frac{{\mathop{H_{{m-1}}\/}\nolimits^{{\prime}}}\!\left(z% \right)}{\mathop{H_{{m-1}}\/}\nolimits\!\left(z\right)},$ $m=1,2,3,\dots$ ,

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : integer and : real
Permalink:: http://dlmf.nist.gov/32.10.E20
Encodings:: TeX, pMML, png

and

32.10.21 $w(z;-m,-2(m+1)^{2})=-2z+\frac{{\mathop{H_{{m}}\/}\nolimits^{{\prime}}}\!\left(% z\right)}{\mathop{H_{{m}}\/}\nolimits\!\left(z\right)},$ $m=0,1,2,\dots$ .

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : integer and : real
Permalink:: http://dlmf.nist.gov/32.10.E21
Encodings:: TeX, pMML, png

If $1+\varepsilon\alpha=0$ , then (32.10.17) has solutions

32.10.22 $w(z)=\begin{cases}\dfrac{2\mathop{\exp\/}\nolimits\!\left(z^{2}\right)}{\sqrt{% \pi}\left(C-i\mathop{\mathrm{erfc}\/}\nolimits\!\left(iz\right)\right)},&% \varepsilon=1,\\ \dfrac{2\mathop{\exp\/}\nolimits\!\left(-z^{2}\right)}{\sqrt{\pi}\left(C-% \mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right)\right)},&\varepsilon=-1,\end% {cases}$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{\exp\/}\nolimits z$ : exponential function, : real, $\varepsilon=\pm 1$ and : constant
Permalink:: http://dlmf.nist.gov/32.10.E22
Encodings:: TeX, pMML, png

where is an arbitrary constant and $\mathop{\mathrm{erfc}\/}\nolimits$ is the complementary error function (§7.2(i)).

For examples and plots see Bassom et al. (1995). For determinantal representations see Forrester and Witte (2001) and Okamoto (1986).

§32.10(v) Fifth Painlevé Equation

Notes:: See Gromak and Lukaševič (1982), Lukaševič (1968), Watanabe (1995); also Gromak et al. (2002, §40).
Keywords:: Painlevé equations, Whittaker functions
Referenced by:: §32.8(v)
Permalink:: http://dlmf.nist.gov/32.10.v

If $\delta\neq 0$ , then as in §32.2(ii) we may set $\delta=-\tfrac{1}{2}$ . $\mbox{P}_{{\mbox{\scriptsize V}}}$ then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff

32.10.23 $a+b+\varepsilon_{3}\gamma=2n+1,$

Symbols:: : integer, , , $\varepsilon_{j}=\pm 1$ and $\gamma$ : arbitrary constant
Referenced by:: §32.10(v)
Permalink:: http://dlmf.nist.gov/32.10.E23
Encodings:: TeX, pMML, png

32.10.24

Symbols:: : integer, and
Permalink:: http://dlmf.nist.gov/32.10.E24
Encodings:: TeX, pMML, png

where $n\in\Integer$ , $a=\varepsilon_{1}\sqrt{2\alpha}$ , and $b=\varepsilon_{2}\sqrt{-2\beta}$ , with $\varepsilon_{j}=\pm 1$ , , independently. In the case when in (32.10.23), the Riccati equation is

32.10.25 $zw^{{\prime}}=aw^{2}+(b-a+\varepsilon_{3}z)w-b.$

Symbols:: : real, , and $\varepsilon_{j}=\pm 1$
Referenced by:: §32.10(v)
Permalink:: http://dlmf.nist.gov/32.10.E25
Encodings:: TeX, pMML, png

If $a\neq 0$ , then (32.10.25) has the solution

32.10.26 $w(z)=-z\phi^{{\prime}}(z)/(a\phi(z)),$

Symbols:: : real, and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E26
Encodings:: TeX, pMML, png

where

32.10.27 $\phi(z)=\frac{C_{1}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(\zeta\right)+C_% {2}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(\zeta\right)}{\zeta^{{(a-b+1)/2% }}}\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{2}\zeta\right),$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\exp\/}\nolimits z$ : exponential function, : real, , , $\phi(z)$ : function, $\kappa$ , $\mu$ and $C_{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.10.E27
Encodings:: TeX, pMML, png

with $\zeta=\varepsilon_{3}z$ , $\kappa=\tfrac{1}{2}(a-b+1)$ , $\mu=\tfrac{1}{2}(a+b)$ , and $C_{1}$ , $C_{2}$ arbitrary constants.

For determinantal representations see Forrester and Witte (2002), Masuda (2004), and Okamoto (1987b).

§32.10(vi) Sixth Painlevé Equation

Notes:: See Fokas and Yortsos (1981), Lukaševič and Yablonskiĭ (1967), Okamoto (1987a); also Gromak et al. (2002, §44).
Keywords:: Painlevé equations, hypergeometric function
Referenced by:: §15.17(i), §32.8(vi)
Permalink:: http://dlmf.nist.gov/32.10.vi

$\mbox{P}_{{\mbox{\scriptsize VI}}}$ has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff

32.10.28

Symbols:: : integer, , , and
Permalink:: http://dlmf.nist.gov/32.10.E28
Encodings:: TeX, pMML, png

where $n\in\Integer$ , $a=\varepsilon_{1}\sqrt{2\alpha}$ , $b=\varepsilon_{2}\sqrt{-2\beta}$ , $c=\varepsilon_{3}\sqrt{2\gamma}$ , and $d=\varepsilon_{4}\sqrt{1-2\delta}$ , with $\varepsilon_{j}=\pm 1$ , , independently. If , then the Riccati equation is

32.10.29 $w^{{\prime}}=\frac{aw^{2}}{z(z-1)}+\frac{(b+c)z-a-c}{z(z-1)}w-\frac{b}{z-1}.$

Symbols:: : real, , and
Referenced by:: §32.10(vi)
Permalink:: http://dlmf.nist.gov/32.10.E29
Encodings:: TeX, pMML, png

If $a\neq 0$ , then (32.10.29) has the solution

32.10.30

$w(z)=\frac{\zeta-1}{a\phi(\zeta)}\frac{d\phi}{d\zeta},$

$\zeta=\frac{1}{1-z},$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real, and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E30
Encodings:: TeX, TeX, pMML, pMML, png, png

where

32.10.31 $\phi(\zeta)=C_{1}\mathop{F\/}\nolimits\!\left(b,-a;b+c;\zeta\right)+C_{2}\zeta% ^{{-b+1-c}}\*\mathop{F\/}\nolimits\!\left(-a-b-c+1,-c+1;2-b-c;\zeta\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, , , , $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.10.E31
Encodings:: TeX, pMML, png

with $C_{1}$ , $C_{2}$ arbitrary constants.

Next, let $\Lambda=\Lambda(u,z)$ be the elliptic function (§§22.15(ii), 23.2(iii)) defined by

32.10.32 $u=\int_{0}^{\Lambda}\frac{dt}{\sqrt{t(t-1)(t-z)}},$

Symbols:: : differential of , $\int$ : integral, : real and $\Lambda(u,z)$ : elliptic function
Permalink:: http://dlmf.nist.gov/32.10.E32
Encodings:: TeX, pMML, png

where the fundamental periods $2\phi_{1}$ and $2\phi_{2}$ are linearly independent functions satisfying the hypergeometric equation

32.10.33 $z(1-z)\frac{{d}^{2}\phi}{{dz}^{2}}+(1-2z)\frac{d\phi}{dz}-\tfrac{1}{4}\phi=0.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E33
Encodings:: TeX, pMML, png

Then $\mbox{P}_{{\mbox{\scriptsize VI}}}$ , with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$ , has the general solution

32.10.34 $w(z;0,0,0,\tfrac{1}{2})=\Lambda(C_{1}\phi_{1}+C_{2}\phi_{2},z),$

Symbols:: : real, $\Lambda(u,z)$ : elliptic function, $\phi_{j}$ : periods and : constant
Referenced by:: §32.10(vi)
Permalink:: http://dlmf.nist.gov/32.10.E34
Encodings:: TeX, pMML, png

with $C_{1}$ , $C_{2}$ arbitrary constants. The solution (32.10.34) is an essentially transcendental function of both constants of integration since $\mbox{P}_{{\mbox{\scriptsize VI}}}$ with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{{\prime}},C)=0$ , with a constant.

For determinantal representations see Forrester and Witte (2004) and Masuda (2004).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 32.9 Other Elementary Solutions 32.11 Asymptotic Approximations for Real Variables

§32.10 Special Function Solutions

Contents

§32.10(i) Introduction

§32.10(ii) Second Painlevé Equation

§32.10(iii) Third Painlevé Equation

§32.10(iv) Fourth Painlevé Equation

§32.10(v) Fifth Painlevé Equation

§32.10(vi) Sixth Painlevé Equation