Abelian functions for trigonal curves of genus three

We develop the theory of generalized Weierstrass s- and P-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the P-functions, a proof that the coefficients of the power series expansion of the s-function are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae. © The Author 2007.