In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions.
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In explicit form they are expressible in terms of Jacobian elliptic functions which are periodic functions ( effectively generalisations of trigonometrical functions ).
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My immediate interests are in special functions, particularly elliptic functions and elliptic integrals, even more specifically, integration in terms of Jacobian elliptic functions.
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The twelve Jacobian elliptic functions are then pq, where each of p and q is a different one of the letters s, c, d, n.
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The configurations correspondingly responsible for higher, i . e . excited, states are periodic instantons defined on a circle of Euclidean time which in explicit form are expressed in terms of Jacobian elliptic functions ( the generalization of trigonometric functions ).
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By starting with the Weierstrass p-function and associating with it a group of doubly periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the existing notation to provide a more systematic approach to the subject.
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