View source for Jacobi theta functions (notational variations)

There are a number of notational systems for the [[Theta function|Jacobi theta functions]]. The notations given in the Wikipedia article define the original function
:<math>
\vartheta_{00}(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)
</math>
which is equivalent to
:<math>
\vartheta_{00}(w, q) = \sum_{n=-\infty}^\infty q^{n^2} w^{2n}
</math>
where <math>q=e^{\pi i\tau}</math> and <math>w=e^{\pi iz}</math>.

However, a similar notation is defined somewhat differently in [[Whittaker and Watson]], p.&nbsp;487:
:<math>
\vartheta_{0,0}(x) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 \pi i n x/a)
</math>
This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define
:<math>
\vartheta_{1,1}(x) = \sum_{n=-\infty}^\infty (-1)^n q^{(n+1/2)^2} \exp (\pi i (2 n + 1) x/a)
</math>
This is a factor of ''i'' off from the definition of <math>\vartheta_{11}</math> 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
:<math>
\vartheta_1(z) = -i \sum_{n=-\infty}^\infty (-1)^n q^{(n+1/2)^2} \exp ((2 n + 1) i z)</math>
:<math>
\vartheta_2(z) = \sum_{n=-\infty}^\infty q^{(n+1/2)^2} \exp ((2 n + 1) i z)</math>
:<math>
\vartheta_3(z) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 n i z)</math>
:<math>
\vartheta_4(z) = \sum_{n=-\infty}^\infty (-1)^n q^{n^2} \exp (2 n i z)</math>

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 <math>\vartheta_j</math>. 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 <math>\vartheta(z)</math> should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of <math>\vartheta(z)</math> is intended.

==References==
* {{AS ref|16.27ff.}}
* {{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 <!-- |url=https://books.google.com/books?id=F7jiBQAAQBAJ |access-date=2016-02-21 --> |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 &theta; functions)''

[[Category:Theta functions]]
[[Category:Elliptic functions]]