Theta Function - Auxiliary Functions

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

\begin{align} \vartheta_{01}(z;\tau)& = \vartheta\!\left(z+{\textstyle\frac{1}{2}};\tau\right)\\ \vartheta_{10}(z;\tau)& = \exp\!\left({\textstyle\frac{1}{4}}\pi i \tau + \pi i z\right) \vartheta\!\left(z + {\textstyle\frac{1}{2}}\tau;\tau\right)\\ \vartheta_{11}(z;\tau)& = \exp\!\left({\textstyle\frac{1}{4}}\pi i \tau + \pi i\!\left(z+{\textstyle \frac{1}{2}}\right)\right)\vartheta\!\left(z+{\textstyle\frac{1}{2}}\tau + {\textstyle\frac{1}{2}};\tau\right). \end{align}

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:

\begin{align} \theta_1(z;q) &= -\vartheta_{11}(z;\tau)\\ \theta_2(z;q) &= \vartheta_{10}(z;\tau)\\ \theta_3(z;q) &= \vartheta_{00}(z;\tau)\\ \theta_4(z;q) &= \vartheta_{01}(z;\tau) \end{align}

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

\vartheta_{00}(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4

which is the Fermat curve of degree four.

Read more about this topic:  Theta Function

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