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

${\vartheta }_{00}(z;\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\exp (\pi i{n}^2\tau +2\pi inz)$

which is equivalent to

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

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:

${\vartheta }_{0,0}(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}\exp (2\pi inx/a)$

This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define

${\vartheta }_{1,1}(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{(n+1/2{)}^2}\exp (\pi i(2n+1)x/a)$

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

${\vartheta }_1(z)=-i\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{(n+1/2{)}^2}\exp ((2n+1)iz)$

${\vartheta }_2(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{(n+1/2{)}^2}\exp ((2n+1)iz)$

${\vartheta }_3(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}\exp (2niz)$

${\vartheta }_4(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{n^2}\exp (2niz)$

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

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• 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)