Theta vocabulary I

Abstract

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions ${\theta }_1$, ${\theta }_2$, ${\theta }_3$, ${\theta }_4$ of one complex variable, presented in a ready-to-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters $\tau$ and $2\tau$.

19 pages, minor corrections

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