# 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

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

which is equivalent to

${\displaystyle \vartheta _{00}(z,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2\pi inz)}$

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

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

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

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

This is a factor of i off from the definition of ${\displaystyle \vartheta _{11}}$ 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

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

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 ${\displaystyle \vartheta _{j}}$. 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 ${\displaystyle \vartheta (z)}$ should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of ${\displaystyle \vartheta (z)}$ is intended.