Higher genus Abelian functions

or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. We investigate the theory of Abelian functions with periodicity properties defined from an associated algebraic curve. A thorough summary of the background material is given, including a synopsis of elliptic function theory, generalisations of the Weierstrass σ and ℘-functions and a literature review. The theory of Abelian functions associated with a tetragonal curve of genus six is con-sidered in detail. Differential equations and addition formula satisfied by the functions are derived and a solution to the Jacobi Inversion Problem is presented. New methods which centre on a series expansion of the σ-function are used and discussions on the large com-putations involved are included. We construct a solution to the KP equation using these functions and outline how a general class of solutions can be generated from a wider class of curves. We proceed to present new approaches used to complete results for the lower genu