There are a number of notational systems for the Jacobi theta functions. The notations given in the Wikipedia article define the original function
ϑ 00 ( z ; τ ) = ∑ n = − ∞ ∞ exp ( π i n 2 τ + 2 π i n z ) which is equivalent to
ϑ 00 ( z , q ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 π i n z ) However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487:
ϑ 0 , 0 ( x ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 π i n x / a ) This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define
ϑ 1 , 1 ( x ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( π i ( 2 n + 1 ) x / a ) This is a factor of i off from the definition of ϑ 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
ϑ 1 ( z ) = − i ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) ϑ 2 ( z ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) ϑ 3 ( z ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 n i z ) ϑ 4 ( z ) = ∑ n = − ∞ ∞ ( − 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 ϑ 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 ϑ ( z ) should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of ϑ ( z ) is intended.
