2008/08/26 by Yu. V. Brezhnev, Brezhnev, Yu. V.
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.0808.3486
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical θ-functions of Jacobi and σ-functions of Weierstrass: ordinary differential equations and series expansions. We also give the extension of canonical θ-functions and consider an application to the sixth Painlevé equation (P6). Picard--Hitchin's general solution of P6 is represented explicitly in a form of logarithmic derivative of a corresponding τ-function (Painlevé's form).