A tau-function solution to the sixth Painleve transcendent

Details A tau-function solution to the sixth Painleve transcendent A tau-function solution to the sixth Painleve transcendent

2010-11-07

arxiv.org/abs/1011.1641v2

Theor. Math. Phys. (2009), 161(3), 1616-1633

Mathematics

Classical Analysis and ODEs

English, LaTeX, 21 pages, 2 figures (2 references corrected)

Scientific paper

We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $\tau$-functions. These functions are expressible 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 P6-equation and the uniformization of algebraic curves and present examples.

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