Auxiliary Functions
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
The auxiliary (or half-period) functions are defined by
This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = exp(πiτ) rather than τ. In Jacobi's notation the θ-functions are written like this:
The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions - notational variations for further discussion.
If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane (sometimes called theta constants.) These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
which is the Fermat curve of degree four.
Read more about this topic: Theta Function
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