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

$\vartheta _{00}(z,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2\pi inz)$

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

$\vartheta _{1}(z)=-i\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp((2n+1)z)$

$\vartheta _{2}(z)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\exp((2n+1)z)$

$\vartheta _{3}(z)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2nz)$

$\vartheta _{4}(z)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\exp(2nz)$

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)