§20.5 Infinite Products and Related Results

§20.5(i) Single Products

 20.5.1 $\theta_{1}\left(z,q\right)=2q^{1/4}\sin z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1-2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$
 20.5.2 $\theta_{2}\left(z,q\right)=2q^{1/4}\cos z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1+2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$
 20.5.3 $\theta_{3}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $n$: integer, $z$: complex and $q$: nome Referenced by: §23.15(ii), §23.17(iii), §23.19 Permalink: http://dlmf.nist.gov/20.5.E3 Encodings: TeX, pMML, png See also: Annotations for 20.5(i), 20.5 and 20
 20.5.4 $\theta_{4}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $n$: integer, $z$: complex and $q$: nome Referenced by: §20.4(i), §20.5(i) Permalink: http://dlmf.nist.gov/20.5.E4 Encodings: TeX, pMML, png See also: Annotations for 20.5(i), 20.5 and 20
 20.5.5 $\theta_{1}\left(z\middle|\tau\right)=\theta_{1}'\left(0\middle|\tau\right)\sin z% \prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z\right)\sin\left(n\pi\tau-z% \right)}{{\sin^{2}}\left(n\pi\tau\right)},$
 20.5.6 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\cos z% \prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z\right)\cos\left(n\pi\tau-z% \right)}{{\cos^{2}}\left(n\pi\tau\right)},$
 20.5.7 $\theta_{3}\left(z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})\pi\tau+z\right)\cos\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\cos^{2}}\left((n-\tfrac{1}{2})\pi\tau\right)},$
 20.5.8 $\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})\pi\tau+z\right)\sin\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\sin^{2}}\left((n-\tfrac{1}{2})\pi\tau\right)}.$

Jacobi’s Triple Product

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

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 $\frac{\theta_{1}'\left(z,q\right)}{\theta_{1}\left(z,q\right)}-\cot z=4\sin% \left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos\left(2z\right)+q% ^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin\left(2nz\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $\cot\NVar{z}$: cotangent function, $\sin\NVar{z}$: sine function, $n$: integer, $z$: complex and $q$: nome A&S Ref: 16.29.1 Referenced by: §20.5(ii), §20.5(ii) Permalink: http://dlmf.nist.gov/20.5.E10 Encodings: TeX, pMML, png See also: Annotations for 20.5(ii), 20.5 and 20
 20.5.11 $\frac{\theta_{2}'\left(z,q\right)}{\theta_{2}\left(z,q\right)}+\tan z=-4\sin% \left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos\left(2z\right)+q% ^{4n}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin\left(2nz\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $\tan\NVar{z}$: tangent function, $n$: integer, $z$: complex and $q$: nome A&S Ref: 16.29.2 Referenced by: §20.5(ii) Permalink: http://dlmf.nist.gov/20.5.E11 Encodings: TeX, pMML, png See also: Annotations for 20.5(ii), 20.5 and 20

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

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

 20.5.12 $\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left(z,q\right)}=-4\sin\left(2z% \right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos\left(2z\right)+q^{4n% -2}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $n$: integer, $z$: complex and $q$: nome A&S Ref: 16.29.3 Permalink: http://dlmf.nist.gov/20.5.E12 Encodings: TeX, pMML, png See also: Annotations for 20.5(ii), 20.5 and 20
 20.5.13 $\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left(z,q\right)}=4\sin\left(2z% \right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos\left(2z\right)+q^{4n% -2}}=4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $n$: integer, $z$: complex and $q$: nome A&S Ref: 16.29.4 Referenced by: §20.5(ii) Permalink: http://dlmf.nist.gov/20.5.E13 Encodings: TeX, pMML, png See also: Annotations for 20.5(ii), 20.5 and 20

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

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)).