Jacobi theta functions (notational variations) explained

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 ^{inf ty} _{n =-inf ty} \exp (\pi i n ^{2} \tau + 2 \pi i n z) which is equivalent to\vartheta _{00} (w, q) = \sum ^{inf ty} _{n =-inf ty} q ^{n ^{2}} w ^{2n} where q = e ^{\pi} and w = e ^{\pi} .

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

\vartheta _{0,0} (x) = \sum ^{inf ty} _{n =-inf ty} 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 ^{inf ty} _{n =-inf ty} ( -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 ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} q ^{(n +1 / 2 ) ^{2}} \exp (( 2 n + 1 ) i z) \vartheta _{2(z)} = \sum ^{inf ty} _{n =-inf ty} q ^{(n +1 / 2 ) ^{2}} \exp (( 2 n + 1 ) i z) \vartheta _{3(z)} = \sum ^{inf ty} _{n =-inf ty} q ^{n ^{2}} \exp ( 2 n i z) \vartheta _{4(z)} = \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} q ^{n ^{2}} \exp ( 2 n i 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

• • Book: Table of Integrals, Series, and Products . Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Scripta Technica, Inc. . 1980 . 4th corrected and enlarged . English . . 0-12-294760-6 . 79027143 . 2016-02-21 --> . Gradshteyn and Ryzhik . 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)