§20.5 Infinite Products and Related Results

20.5.1		$${\theta }_1(z,q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome Referenced by: §20.4(i), §20.5(i), §23.17(iii) Permalink: http://dlmf.nist.gov/20.5.E1 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

20.5.2		$${\theta }_2(z,q)=2q^{1/4}\cos z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.5.E2 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

20.5.3		$${\theta }_3(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right),$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: 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 Ch.20

20.5.4		$${\theta }_4(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right).$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: 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 Ch.20

20.5.5		$${\theta }_1\left(z\right|\tau )={\theta }_1^{\prime }\left(0\right|\tau )\sin z\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\pi \tau +z\right)\sin \left(n\pi \tau -z\right)}{{\sin }^2\left(n\pi \tau \right)},$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $z$: complex and $\tau$: lattice parameter Referenced by: §20.5(i), §20.5(iii), §20.7(iv) Permalink: http://dlmf.nist.gov/20.5.E5 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

20.5.6		$${\theta }_2\left(z\right|\tau )={\theta }_2\left(0\right|\tau )\cos z\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\pi \tau +z\right)\cos \left(n\pi \tau -z\right)}{{\cos }^2\left(n\pi \tau \right)},$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.5.E6 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

20.5.7		$${\theta }_3\left(z\right|\tau )={\theta }_3\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left((n-\frac{1}{2})\pi \tau +z\right)\cos \left((n-\frac{1}{2})\pi \tau -z\right)}{{\cos }^2\left((n-\frac{1}{2})\pi \tau \right)},$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.5.E7 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

20.5.8		$${\theta }_4\left(z\right|\tau )={\theta }_4\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left((n-\frac{1}{2})\pi \tau +z\right)\sin \left((n-\frac{1}{2})\pi \tau -z\right)}{{\sin }^2\left((n-\frac{1}{2})\pi \tau \right)}.$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $z$: complex and $\tau$: lattice parameter Referenced by: §20.5(iii), §20.7(iv) Permalink: http://dlmf.nist.gov/20.5.E8 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5 and Ch.20

When ,

20.5.10		$$\begin{array}{ll}\frac{{\theta }_1^{\prime }(z,q)}{{\theta }_1(z,q)}-\cot z& =4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-2q^{2n}\cos \left(2z\right)+q^{4n}}\\ & =4\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right),\end{array}$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: 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 Ch.20

20.5.11		$$\begin{array}{ll}\frac{{\theta }_2^{\prime }(z,q)}{{\theta }_2(z,q)}+\tan z& =-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1+2q^{2n}\cos \left(2z\right)+q^{4n}}\\ & =4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right).\end{array}$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: 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 Ch.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 ,

20.5.12		$$\begin{array}{ll}\frac{{\theta }_3^{\prime }(z,q)}{{\theta }_3(z,q)}& =-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}\\ & =4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^n}{1-q^{2n}}\sin \left(2nz\right),\end{array}$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: 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 Ch.20

20.5.13		$$\begin{array}{ll}\frac{{\theta }_4^{\prime }(z,q)}{{\theta }_4(z,q)}& =4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}\\ & =4\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^{2n}}\sin \left(2nz\right).\end{array}$$
ⓘ Symbols: ${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: 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 Ch.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.

20.5.14		${\theta }_1\left(z\right|\tau )$	$=z{\theta }_1^{\prime }\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \prod _{n=-N}^N}\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \prod _{\begin{array}{c}\hfill m=-M\hfill \\ \hfill \left|m\right|+\left|n\right|\ne 0\hfill \end{array}}^M}\left(1+{\displaystyle \frac{z}{(m+n\tau )\pi }}\right),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.5.E14 Encodings: TeX, pMML, png See also: Annotations for §20.5(iii), §20.5 and Ch.20
20.5.15		${\theta }_2\left(z\right|\tau )$	$={\theta }_2\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \prod _{n=-N}^N}\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \prod _{m=1-M}^M}\left(1+{\displaystyle \frac{z}{(m-\frac{1}{2}+n\tau )\pi }}\right),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.5.E15 Encodings: TeX, pMML, png See also: Annotations for §20.5(iii), §20.5 and Ch.20
20.5.16		${\theta }_3\left(z\right|\tau )$	$={\theta }_3\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \prod _{n=1-N}^N}\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \prod _{m=1-M}^M}\left(1+{\displaystyle \frac{z}{(m-\frac{1}{2}+(n-\frac{1}{2})\tau )\pi }}\right),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.5.E16 Encodings: TeX, pMML, png See also: Annotations for §20.5(iii), §20.5 and Ch.20
20.5.17		${\theta }_4\left(z\right|\tau )$	$={\theta }_4\left(0\right|\tau )\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \prod _{n=1-N}^N}\underset{M\to \mathrm{\infty }}{lim}{\displaystyle \prod _{m=-M}^M}\left(1+{\displaystyle \frac{z}{(m+(n-\frac{1}{2})\tau )\pi }}\right).$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.5.E17 Encodings: TeX, pMML, png See also: Annotations for §20.5(iii), §20.5 and Ch.20

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)).