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A τ-function solution of the sixth painlevé transcendent

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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 τ-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.

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Correspondence to Yu. V. Brezhnev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 161, No. 3, pp. 346–366, December, 2009.

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Brezhnev, Y.V. A τ-function solution of the sixth painlevé transcendent. Theor Math Phys 161, 1616–1633 (2009). https://doi.org/10.1007/s11232-009-0150-z

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Keywords

  • Painlevé VI equation
  • elliptic function
  • theta function
  • uniformization
  • automorphic function