A tau-function solution to the sixth Painleve transcendent

Abstract

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.

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

References in corpus (1)

• The sixth Painleve transcendent and uniformization of algebraic curves