This article is cited in 12 papers

A $\tau$-function solution of the sixth Painlevé transcendent

Yu. V. Brezhnev

Tomsk State University, Tomsk, Russia

Abstract: 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.

Keywords: Painlevé VI equation, elliptic function, theta function, uniformization, automorphic function.

Received: 04.03.2009
Revised: 08.04.2009

DOI: 10.4213/tmf6446

 English version: Theoretical and Mathematical Physics, 2009, 161:3, 1616–1633

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