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We consider specific classes of algebraic curves and define a set of abelian function upon their Jacobian.We do this with the aim of generalising the theory of the Weier strass (ω)-function and the associated theory of elliptic curves.A variety of methods are employed to find addition formula, PDEs, power series expansions and other properties that these functions satisfy.This research lies within the field of algebraic geometry, and has applications in non-linear PDEs and Soliton theory.