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Jacobi theta functions (notational variations)

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There are a number of notational systems for the Jacobi theta functions. The notations given in the Misplaced Pages 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 Misplaced Pages 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)}
ϑ 4 ( z ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 exp ⁡ ( 2 n i z ) {\displaystyle \vartheta _{4}(z)=\sum _{n=-\infty }^{\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 ϑ j {\displaystyle \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 ϑ ( z ) {\displaystyle \vartheta (z)} should not be assumed to have any particular definition. It is incumbent upon the author to state what definition of ϑ ( z ) {\displaystyle \vartheta (z)} is intended.

References

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