Geodesic
A $\tau$-function solution of the sixth Painlevé transcendent
Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 346-366 Cet article a éte moissonné depuis la source Math-Net.Ru

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We represent and analyze the general solution of the sixth Painlevé transcendent $\mathcal P_6$ in the Picard–Hitchin–Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of $\tau$-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $\mathcal P_6$ equation and the uniformization of algebraic curves and present examples.
Mots-clés : Painlevé VI equation
Keywords: elliptic function, theta function, uniformization, automorphic function. 
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Yu. V. Brezhnev. A $\tau$-function solution of the sixth Painlevé transcendent. Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 346-366. http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a4/

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