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.

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 16.27ff.". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (1980). "8.18.". In Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (4th corrected and enlarged ed.). Academic Press, Inc. ISBN 0-12-294760-6. LCCN 79027143.
• 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)