Theta vocabulary I

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $θ_1$, $θ_2$, $θ_3$, $θ_4$ 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 $τ$ and $2τ$.

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