# PhD Defence: Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities

**Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities**

The main objects of this thesis are the Jacobi theta functions $\theta_j(z|\tau)$, $j=1,\dots,4$, and the classes of problems we consider are algebraic relations between them. In the past centuries researchers, including mathematicians, physicists, etc., have been using some difficult arithmetic manipulations to prove even basic theta function identities by hand, a tedious (perhaps unfeasible) task for more complicated identities.

In this thesis we present computer algebra algorithms to deal with several general classes of theta function identities in computer-assisted manner. One essential mathematical tool used for developing these algorithms is complex analysis.

Our algorithmic approaches can be used to prove identities from very general function classes within a few minutes; in addition, we can also discover identities from such classes in computer-assisted way. We have implemented the algorithms into a Mathematica package "ThetaFunctions".

As a by-product, relations (old and new) involving the Weierstrass elliptic function are found. Moreover, our algorithmic approaches can be extended further to other classes of identities, for example a substantial amount of identities in Ramanujan's lost notebooks and in other research monographs and papers.