# §32.9 Other Elementary Solutions

## §32.9(i) Third Painlevé Equation

Elementary nonrational solutions of $\mbox{P}_{\mbox{\scriptsize III}}$ are

 32.9.1 $w(z;\mu,0,0,-\mu\kappa^{3})=\kappa z^{1/3},$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E1 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32
 32.9.2 $w(z;0,-2\kappa,0,4\kappa\mu-\lambda^{2})=z(\kappa(\ln z)^{2}+\lambda\ln z+\mu),$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $z$: real, $\kappa$: constant, $\lambda$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E2 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32
 32.9.3 $w(z;-\nu^{2}\lambda,0,\nu^{2}(\lambda^{2}-4\kappa\mu),0)=\dfrac{z^{\nu-1}}{% \kappa z^{2\nu}+\lambda z^{\nu}+\mu},$ ⓘ Symbols: $z$: real, $\kappa$: constant, $\lambda$: constant, $\mu$: constant and $\nu$: constant Permalink: http://dlmf.nist.gov/32.9.E3 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32

with $\kappa$, $\lambda$, $\mu$, and $\nu$ arbitrary constants.

In the case $\gamma=0$ and $\alpha\delta\neq 0$ we assume, as in §32.2(ii), $\alpha=1$ and $\delta=-1$. Then $\mbox{P}_{\mbox{\scriptsize III}}$ has algebraic solutions iff

 32.9.4 $\beta=2n,$ ⓘ Symbols: $n$: integer and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.9.E4 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32

with $n\in\mathbb{Z}$. These are rational solutions in $\zeta=z^{1/3}$ of the form

 32.9.5 $w(z)=\ifrac{P_{n^{2}+1}(\zeta)}{Q_{n^{2}}(\zeta)},$ ⓘ Symbols: $n$: integer, $z$: real, $P_{n^{2}+1}(\zeta)$: polynomials and $Q_{n^{2}}(\zeta)$: polynomials Permalink: http://dlmf.nist.gov/32.9.E5 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32

where $P_{n^{2}+1}(\zeta)$ and $Q_{n^{2}}(\zeta)$ are polynomials of degrees $n^{2}+1$ and $n^{2}$, respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when $\delta=0$ and $\beta\gamma\neq 0$.

$\mbox{P}_{\mbox{\scriptsize III}}$ with $\beta=\delta=0$ has a first integral

 32.9.6 $z^{2}(w^{\prime})^{2}+2zww^{\prime}=(C+2\alpha zw+\gamma z^{2}w^{2})w^{2},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\gamma$: arbitrary constant and $C$: constant Permalink: http://dlmf.nist.gov/32.9.E6 Encodings: TeX, pMML, png See also: Annotations for 32.9(i), 32.9 and 32

with $C$ an arbitrary constant, which is solvable by quadrature. A similar result holds when $\alpha=\gamma=0$. $\mbox{P}_{\mbox{\scriptsize III}}$ with $\alpha=\beta=\gamma=\delta=0$, has the general solution $w(z)=Cz^{\mu}$, with $C$ and $\mu$ arbitrary constants.

## §32.9(ii) Fifth Painlevé Equation

Elementary nonrational solutions of $\mbox{P}_{\mbox{\scriptsize V}}$ are

 32.9.7 $w(z;\mu,-\tfrac{1}{8},-\mu\kappa^{2},0)=1+\kappa z^{1/2},$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E7 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32
 32.9.8 $w(z;0,0,\mu,-\tfrac{1}{2}\mu^{2})=\kappa\exp\left(\mu z\right),$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E8 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32

with $\kappa$ and $\mu$ arbitrary constants.

$\mbox{P}_{\mbox{\scriptsize V}}$, with $\delta=0$, has algebraic solutions if either

 32.9.9 $\mspace{17.0mu }(\alpha,\beta,\gamma)=(\tfrac{1}{2}\mu^{2},-\tfrac{1}{8}(2n-1)% ^{2},-1),$ ⓘ Symbols: $n$: integer, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E9 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32

or

 32.9.10 $\mspace{5.0mu }(\alpha,\beta,\gamma)=(\tfrac{1}{8}(2n-1)^{2},-\tfrac{1}{2}\mu^% {2},1),$ ⓘ Symbols: $n$: integer, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E10 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32

with $n\in\mathbb{Z}$ and $\mu$ arbitrary. These are rational solutions in $\zeta=z^{1/2}$ of the form

 32.9.11 $w(z)=\ifrac{P_{n^{2}-n+1}(\zeta)}{Q_{n^{2}-n}(\zeta)},$ ⓘ Symbols: $n$: integer, $z$: real, $P_{n^{2}+1}(\zeta)$: polynomials and $Q_{n^{2}}(\zeta)$: polynomials Permalink: http://dlmf.nist.gov/32.9.E11 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32

where $P_{n^{2}-n+1}(\zeta)$ and $Q_{n^{2}-n}(\zeta)$ are polynomials of degrees $n^{2}-n+1$ and $n^{2}-n$, respectively, with no common zeros.

$\mbox{P}_{\mbox{\scriptsize V}}$, with $\gamma=\delta=0$, has a first integral

 32.9.12 $z^{2}(w^{\prime})^{2}=(w-1)^{2}(2\alpha w^{2}+Cw-2\beta),$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and $C$: constant Permalink: http://dlmf.nist.gov/32.9.E12 Encodings: TeX, pMML, png See also: Annotations for 32.9(ii), 32.9 and 32

with $C$ an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). $\mbox{P}_{\mbox{\scriptsize V}}$, with $\alpha=\beta=0$ and $\gamma^{2}+2\delta=0$, has solutions $w(z)=C\exp\left(\pm\sqrt{-2\delta}z\right)$, with $C$ an arbitrary constant.

## §32.9(iii) Sixth Painlevé Equation

An elementary algebraic solution of $\mbox{P}_{\mbox{\scriptsize VI}}$ is

 32.9.13 $w(z;\tfrac{1}{2}\kappa^{2},-\tfrac{1}{2}\kappa^{2},\tfrac{1}{2}\mu^{2},\tfrac{% 1}{2}(1-\mu^{2}))=z^{1/2},$ ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E13 Encodings: TeX, pMML, png See also: Annotations for 32.9(iii), 32.9 and 32

with $\kappa$ and $\mu$ arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\beta=\gamma=0$, $\delta=\tfrac{1}{2}$. For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b).