Utilisateur:Robert Maurice Conte/Brouillon

Fonctions de Painlevé

Painlevé a fait des travaux éminents

Définitions

${\frac {d^{2}U}{dX^{2}}}-{\frac {(2\omega )^{3}}{\pi ^{2}}}\sum _{j=\infty ,0,1,x}\theta _{j}^{2}\wp '(2\omega U+\omega _{j},g_{2},g_{3})=0,$

${\begin{aligned}u''&={\frac {1}{2}}\left[{\frac {1}{u}}+{\frac {1}{u-1}}+{\frac {1}{u-x}}\right]{u'}^{2}-\left[{\frac {1}{x}}+{\frac {1}{x-1}}+{\frac {1}{u-x}}\right]u'\\&+{\frac {u(u-1)(u-x)}{2x^{2}(x-1)^{2}}}\left[\theta _{\infty }^{2}-\theta _{0}^{2}{\frac {x}{u^{2}}}+\theta _{1}^{2}{\frac {x-1}{(u-1)^{2}}}+(1-\theta _{x}^{2}){\frac {x(x-1)}{(u-x)^{2}}}\right]\end{aligned}}$

Coalescence

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