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 ) {\displaystyle \vartheta _{00}(z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz)}
which is equivalent to
ϑ 00 ( w , q ) = ∑ n = − ∞ ∞ q n 2 w 2 n {\displaystyle \vartheta _{00}(w,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}w^{2n}}
where q = e π i τ {\displaystyle q=e^{\pi i\tau }} and w = e π i z {\displaystyle w=e^{\pi iz}} . 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 ) {\displaystyle \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
ϑ 1 , 1 ( x ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( π i ( 2 n + 1 ) x / a ) {\displaystyle \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 ϑ 11 {\displaystyle \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
ϑ 1 ( z ) = − i ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) {\displaystyle \vartheta _{1}(z)=-i\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}\exp((2n+1)iz)}
ϑ 2 ( z ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) {\displaystyle \vartheta _{2}(z)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\exp((2n+1)iz)}
ϑ 3 ( z ) = ∑ n = − ∞ ∞ q n 2 exp ( 2 n i z ) {\displaystyle \vartheta _{3}(z)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\exp(2niz)}
… excerpt ends here. Continue reading the full article.
