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

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

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

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.

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)