Jacobi theta functions (notational variations)
# Jacobi theta functions (notational variations)

There are a number of notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function

    $\vartheta _{00}(z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz)$

which is equivalent to

    $\vartheta _{00}(w,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}w^{2n}$

where $q=e^{\pi i\tau }$ and $w=e^{\pi iz}$.

However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487:

    $\vartheta _{0,0}(x)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2\pi inx/a)$

This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define

    $\vartheta _{1,1}(x)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp(\pi i(2n+1)x/a)$

This is a factor of i off from the definition of $\vartheta _{11}$ as defined in the Wikipedia article. These definitions can be made at least proportional by x = za, but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which

    $\vartheta _{1}(z)=-i\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp((2n+1)iz)$
    $\vartheta _{2}(z)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\exp((2n+1)iz)$
    $\vartheta _{3}(z)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2niz)$
    $\vartheta _{4}(z)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\exp(2niz)$

Note that there is no factor of π in the argument as in the previous definitions.

Whittaker and Watson refer to still other definitions of $\vartheta _{j}$. The warning in Abramowitz and Stegun, "There is a bewildering variety of notations...in consulting books caution should be exercised," may be viewed as an understatement. In any expression, an occurrence of $\vartheta (z)$ should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of $\vartheta (z)$ is intended.

