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
:$\vartheta_(z; \tau) = \sum_^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)$
which is equivalent to
:$\vartheta_(w, q) = \sum_^\infty q^ w^$
where $q=e^$ and $w=e^$.

However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487:
:$\vartheta_(x) = \sum_^\infty q^ \exp (2 \pi i n x/a)$
This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define
:$\vartheta_(x) = \sum_^\infty (-1)^n q^ \exp (\pi i (2 n + 1) x/a)$
This is a factor of ''i'' off from the definition of $\vartheta_$ 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
:$\vartheta_1(z) = -i \sum_^\infty (-1)^n q^ \exp ((2 n + 1) i z)$
:$\vartheta_2(z) = \sum_^\infty q^ \exp ((2 n + 1) i z)$
:$\vartheta_3(z) = \sum_^\infty q^ \exp (2 n i z)$
:$\vartheta_4(z) = \sum_^\infty (-1)^n q^ \exp (2 n i z)$

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.

References

*
* {{cite book , title=Table of Integrals, Series, and Products , author-first1=Izrail Solomonovich , author-last1=Gradshteyn , author-link1=Izrail Solomonovich Gradshteyn , author-first2=Iosif Moiseevich , author-last2=Ryzhik , author-link2=Iosif Moiseevich Ryzhik , author-first3=Yuri Veniaminovich , author-last3=Geronimus , author-link3=Yuri Veniaminovich Geronimus , author-first4=Michail Yulyevich , author-last4=Tseytlin , author-link4=Michail Yulyevich Tseytlin , editor-first1=Alan , editor-last1=Jeffrey , translator=Scripta Technica, Inc. , date=1980 , edition=4th corrected and enlarged , language=English , publisher= Academic Press, Inc. , isbn=0-12-294760-6 , lccn=79027143 , title-link=Gradshteyn and Ryzhik , chapter=8.18.
* E. T. Whittaker and G. N. Watson, ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1927. ''(See chapter XXI for the history of Jacobi's θ functions)''

Theta functions
Elliptic functions