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

[math]\displaystyle{ \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]\displaystyle{ \vartheta_{00}(w, q) = \sum_{n=-\infty}^\infty q^{n^2} w^{2n} }[/math]

where [math]\displaystyle{ q=e^{\pi i\tau} }[/math] and [math]\displaystyle{ w=e^{\pi iz} }[/math].

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

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \vartheta_1(z) = -i \sum_{n=-\infty}^\infty (-1)^n q^{(n+1/2)^2} \exp ((2 n + 1) i z) }[/math]

[math]\displaystyle{ \vartheta_2(z) = \sum_{n=-\infty}^\infty q^{(n+1/2)^2} \exp ((2 n + 1) i z) }[/math]

[math]\displaystyle{ \vartheta_3(z) = \sum_{n=-\infty}^\infty q^{n^2} \exp (2 n i z) }[/math]

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

References

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 16.27ff.". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 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. LCCN 65-12253. ISBN 978-0-486-61272-0. 
• "8.18." (in English). Table of Integrals, Series, and Products (4th corrected and enlarged ed.). Academic Press, Inc.. 1980. ISBN 0-12-294760-6. 
• 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)