# §20.5 Infinite Products and Related Results

## §20.5(i) Single Products

 20.5.1 $\mathop{\theta_{1}\/}\nolimits\!\left(z,q\right)=2q^{1/4}\mathop{\sin\/}% \nolimits z\prod\limits_{n=1}^{\infty}{\left(1-q^{2n}\right)}{\left(1-2q^{2n}% \mathop{\cos\/}\nolimits\!\left(2z\right)+q^{4n}\right)},$
 20.5.2 $\mathop{\theta_{2}\/}\nolimits\!\left(z,q\right)=2q^{1/4}\mathop{\cos\/}% \nolimits z\prod\limits_{n=1}^{\infty}{\left(1-q^{2n}\right)}{\left(1+2q^{2n}% \mathop{\cos\/}\nolimits\!\left(2z\right)+q^{4n}\right)},$
 20.5.3 $\mathop{\theta_{3}\/}\nolimits\!\left(z,q\right)=\prod\limits_{n=1}^{\infty}% \left(1-q^{2n}\right)\left(1+2q^{2n-1}\mathop{\cos\/}\nolimits\!\left(2z\right% )+q^{4n-2}\right),$
 20.5.4 $\mathop{\theta_{4}\/}\nolimits\!\left(z,q\right)=\prod\limits_{n=1}^{\infty}% \left(1-q^{2n}\right)\left(1-2q^{2n-1}\mathop{\cos\/}\nolimits\!\left(2z\right% )+q^{4n-2}\right).$
 20.5.5 $\mathop{\theta_{1}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{1}% \/}\nolimits'\!\left(0\middle|\tau\right)\mathop{\sin\/}\nolimits z\prod_{n=1}% ^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(n\pi\tau+z\right)\mathop{\sin\/% }\nolimits\!\left(n\pi\tau-z\right)}{{\mathop{\sin\/}\nolimits^{2}}\!\left(n% \pi\tau\right)},$
 20.5.6 $\mathop{\theta_{2}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{2}% \/}\nolimits\!\left(0\middle|\tau\right)\mathop{\cos\/}\nolimits z\prod_{n=1}^% {\infty}\frac{\mathop{\cos\/}\nolimits\!\left(n\pi\tau+z\right)\mathop{\cos\/}% \nolimits\!\left(n\pi\tau-z\right)}{{\mathop{\cos\/}\nolimits^{2}}\!\left(n\pi% \tau\right)},$
 20.5.7 $\mathop{\theta_{3}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{3}% \/}\nolimits\!\left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\mathop{\cos% \/}\nolimits\!\left((n-\tfrac{1}{2})\pi\tau+z\right)\mathop{\cos\/}\nolimits\!% \left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\mathop{\cos\/}\nolimits^{2}}\!\left(% (n-\tfrac{1}{2})\pi\tau\right)},$
 20.5.8 $\mathop{\theta_{4}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{4}% \/}\nolimits\!\left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\mathop{\sin% \/}\nolimits\!\left((n-\tfrac{1}{2})\pi\tau+z\right)\mathop{\sin\/}\nolimits\!% \left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\mathop{\sin\/}\nolimits^{2}}\!\left(% (n-\tfrac{1}{2})\pi\tau\right)}.$

### Jacobi’s Triple Product

 20.5.9 $\mathop{\theta_{3}\/}\nolimits\!\left(\pi z\middle|\tau\right)=\sum_{n=-\infty% }^{\infty}p^{2n}q^{n^{2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right),$

where $p=e^{i\pi z}$, $q=e^{i\pi\tau}$.

## §20.5(ii) Logarithmic Derivatives

When $\left|\Im{z}\right|<\pi\Im{\tau}$,

 20.5.10 $\frac{\mathop{\theta_{1}\/}\nolimits'\!\left(z,q\right)}{\mathop{\theta_{1}\/}% \nolimits\!\left(z,q\right)}-\mathop{\cot\/}\nolimits z=4\mathop{\sin\/}% \nolimits\!\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\mathop{% \cos\/}\nolimits\!\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-% q^{2n}}\mathop{\sin\/}\nolimits\!\left(2nz\right),$
 20.5.11 $\frac{\mathop{\theta_{2}\/}\nolimits'\!\left(z,q\right)}{\mathop{\theta_{2}\/}% \nolimits\!\left(z,q\right)}+\mathop{\tan\/}\nolimits z=-4\mathop{\sin\/}% \nolimits\!\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\mathop{% \cos\/}\nolimits\!\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^% {2n}}{1-q^{2n}}\mathop{\sin\/}\nolimits\!\left(2nz\right).$

The left-hand sides of (20.5.10) and (20.5.11) are replaced by their limiting values when $\mathop{\cot\/}\nolimits z$ or $\mathop{\tan\/}\nolimits z$ are undefined.

When $\left|\Im{z}\right|<\tfrac{1}{2}\pi\Im{\tau}$,

 20.5.12 $\frac{\mathop{\theta_{3}\/}\nolimits'\!\left(z,q\right)}{\mathop{\theta_{3}\/}% \nolimits\!\left(z,q\right)}=-4\mathop{\sin\/}\nolimits\!\left(2z\right)\sum_{% n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\mathop{\cos\/}\nolimits\!\left(2z% \right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\mathop{% \sin\/}\nolimits\!\left(2nz\right),$
 20.5.13 $\frac{\mathop{\theta_{4}\/}\nolimits'\!\left(z,q\right)}{\mathop{\theta_{4}\/}% \nolimits\!\left(z,q\right)}=4\mathop{\sin\/}\nolimits\!\left(2z\right)\sum_{n% =1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\mathop{\cos\/}\nolimits\!\left(2z% \right)+q^{4n-2}}=4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\mathop{\sin\/}% \nolimits\!\left(2nz\right).$

With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the $z$-plane.

## §20.5(iii) Double Products

 20.5.14 $\displaystyle\mathop{\theta_{1}\/}\nolimits\!\left(z\middle|\tau\right)$ $\displaystyle=z\mathop{\theta_{1}\/}\nolimits'\!\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{\begin{subarray}{c}m=% -M\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),$ 20.5.15 $\displaystyle\mathop{\theta_{2}\/}\nolimits\!\left(z\middle|\tau\right)$ $\displaystyle=\mathop{\theta_{2}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim% _{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z% }{(m-\tfrac{1}{2}+n\tau)\pi}\right),$ 20.5.16 $\displaystyle\mathop{\theta_{3}\/}\nolimits\!\left(z\middle|\tau\right)$ $\displaystyle=\mathop{\theta_{3}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim% _{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{% z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}\right),$ 20.5.17 $\displaystyle\mathop{\theta_{4}\/}\nolimits\!\left(z\middle|\tau\right)$ $\displaystyle=\mathop{\theta_{4}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim% _{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+\frac{z% }{(m+(n-\tfrac{1}{2})\tau)\pi}\right).$

These double products are not absolutely convergent; hence the order of the limits is important. The order shown is in accordance with the Eisenstein convention (Walker (1996, §0.3)).