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:$vartheta\_\{00\}(z;\; au)\; =\; sum\_\{n=-infty\}^infty\; exp\; (pi\; i\; n^2\; au\; +\; 2\; pi\; i\; n\; z)$which is equivalent to:$vartheta\_\{00\}(z,\; q)\; =\; sum\_\{n=-infty\}^infty\; q^\{n^2\}\; exp\; (2\; pi\; i\; n\; z)$However, a similar notation is defined somewhat differently in Whittaker and Watson, p.487::$vartheta\_\{0,0\}(x)\; =\; sum\_\{n=-infty\}^infty\; q^\{n^2\}\; exp\; (2\; pi\; i\; n\; x/a)$This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define:$vartheta\_\{1,1\}(x)\; =\; sum\_\{n=-infty\}^infty\; (-1)^n\; q^\{(n+1/2)^2\}\; exp\; (pi\; i\; (2\; n\; +\; 1)\; x/a)$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:$vartheta\_1(z)\; =\; -i\; sum\_\{n=-infty\}^infty\; (-1)^n\; q^\{(n+1/2)^2\}\; exp\; ((2\; n\; +\; 1)\; z)$:$vartheta\_2(z)\; =\; sum\_\{n=-infty\}^infty\; q^\{(n+1/2)^2\}\; exp\; ((2\; n\; +\; 1)\; z)$:$vartheta\_3(z)\; =\; sum\_\{n=-infty\}^infty\; q^\{n^2\}\; exp\; (2\; n\; z)$:$vartheta\_4(z)\; =\; sum\_\{n=-infty\}^infty\; (-1)^n\; q^\{n^2\}\; exp\; (2\; n\; z)$

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

* Milton Abramowitz and Irene A. Stegun, "Handbook of Mathematical Functions", (1964) Dover Publications, New York. ISBN 0-486-61272-4. "(See section 16.27ff.)"
* I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Functions, and Products", (1980) Academic Press, London. ISBN 0-12-294760-6. "(See section 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)"