Painlevé transcendents

Definition

Let $\alpha, \beta, \gamma, \delta$ be complex constants, and let $y$ be a function of a complex variable $t$. The six Painlevé equations are the ordinary differential equations:

1. $\mathrm{P_I}$: $y'' = 6y^2 + t$;
2. $\mathrm{P_{II}}$: $y'' = 2y^3 + ty + \alpha$;
3. $\mathrm{P_{III}}$: $y'' = \frac{(y')^2}{y} - \frac{y'}{t} + \frac{\alpha y^2 + \beta}{t} + \gamma y^3 + \frac{\delta}{y}$;
4. $\mathrm{P_{IV}}$: $y'' = \frac{(y')^2}{2y} + \frac32 y^3 + 4ty^2 + 2(t^2 - \alpha) y + \frac{\beta}{y}$;
5. $\mathrm{P_V}$: $y'' = \left( \frac{1}{2y} + \frac{1}{y - 1} \right) (y')^2 - \frac{y'}{t} + \frac{(y - 1)^2}{t^2} \left( \alpha y + \frac{\beta}{y} \right) + \frac{\gamma y}{t} + \frac{\delta y (y + 1)}{y - 1}$;
6. $\mathrm{P_{VI}}$: $y'' = \frac12 \left( \frac1y + \frac{1}{y - 1} + \frac{1}{y - t} \right) (y')^2 - \left( \frac1t + \frac{1}{t - 1} + \frac{1}{y - t} \right) y' + \frac{y (y - 1)(y - t)}{t^2 (t - 1)^2} \left( \alpha + \frac{\beta t}{y^2} + \frac{\gamma (t - 1)}{(y - 1)^2} + \frac{\delta t (t - 1)}{(y - t)^2} \right)$.

A Painlevé transcendent is a solution of one of these six equations.