20 Theta Functions Properties 20.4 Values at

z

§20.5 Infinite Products and Related Results

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Keywords:: infinite products, theta functions
Referenced by:: §17.3(iv)
Permalink:: http://dlmf.nist.gov/20.5
See also:: Annotations for Ch.20

§20.5(i) Single Products

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Notes:: See Lawden (1989, p. 15) for (20.5.1)–(20.5.4), and Walker (1996, pp. 87–92) for (20.5.5)–(20.5.9).
Permalink:: http://dlmf.nist.gov/20.5.i
See also:: Annotations for §20.5 and Ch.20

20.5.1		$\theta_{1}\left(z,q\right)=2q^{1/4}\sin z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1-2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$
ⓘ 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 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		$\theta_{2}\left(z,q\right)=2q^{1/4}\cos z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1+2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : 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		$\theta_{3}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right),$
ⓘ 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 Ch.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 Ch.20

20.5.5		$\theta_{1}\left(z\middle\|\tau\right)=\theta_{1}'\left(0\middle\|\tau\right)\sin z% \prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z\right)\sin\left(n\pi\tau-z% \right)}{{\sin}^{2}\left(n\pi\tau\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : 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		$\theta_{2}\left(z\middle\|\tau\right)=\theta_{2}\left(0\middle\|\tau\right)\cos z% \prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z\right)\cos\left(n\pi\tau-z% \right)}{{\cos}^{2}\left(n\pi\tau\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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		$\theta_{3}\left(z\middle\|\tau\right)=\theta_{3}\left(0\middle\|\tau\right)\prod% _{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})\pi\tau+z\right)\cos\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\cos}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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		$\theta_{4}\left(z\middle\|\tau\right)=\theta_{4}\left(0\middle\|\tau\right)\prod% _{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})\pi\tau+z\right)\sin\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\sin}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : 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

Jacobi’s Triple Product

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Keywords:: Jacobi’s triple product, theta functions
See also:: Annotations for §20.5(i), §20.5 and Ch.20

20.5.9		$\theta_{3}\left(\pi z\middle\|\tau\right)=\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{% 2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome Referenced by: §17.8, §20.12(i), §20.5(i) Permalink: http://dlmf.nist.gov/20.5.E9 Encodings: TeX, pMML, png See also: Annotations for §20.5(i), §20.5(i), §20.5 and Ch.20

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

§20.5(ii) Logarithmic Derivatives

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Keywords:: derivatives, theta functions
Notes:: See Lawden (1989, p. 22), Whittaker and Watson (1927, p. 471), and Bellman (1961, p. 44).
Permalink:: http://dlmf.nist.gov/20.5.ii
See also:: Annotations for §20.5 and Ch.20

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

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 Ch.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 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(iii) Double Products

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Keywords:: double products, infinite products, theta functions
Notes:: These relations follow from equations (20.5.5) – (20.5.8) by use of the infinite products for the sine and cosine (§4.22).
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/20.5.iii
See also:: Annotations for §20.5 and Ch.20

20.5.14	$\displaystyle\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=z\theta_{1}'\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{% n=-N}^{N}\lim_{M\to\infty}\prod_{\begin{subarray}{c}m=-M\\ \left\|m\right\|+\left\|n\right\|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{2}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% -N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n% \tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% 1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-% \tfrac{1}{2})\tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% 1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})% \tau)\pi}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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)).

20.4 Values at

z

= 0 20.6 Power Series

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20.5.14	$\displaystyle\theta_{1}\left(z\middle\|\tau\right)$	$\displaystyle=z\theta_{1}'\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{% n=-N}^{N}\lim_{M\to\infty}\prod_{\begin{subarray}{c}m=-M\\ \left\|m\right\|+\left\|n\right\|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{2}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{2}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% -N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n% \tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{3}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% 1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-% \tfrac{1}{2})\tau)\pi}\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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	$\displaystyle\theta_{4}\left(z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(0\middle\|\tau\right)\*\lim_{N\to\infty}\prod_{n=% 1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})% \tau)\pi}\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : 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

§20.5 Infinite Products and Related Results

Contents

§20.5(i) Single Products

Jacobi’s Triple Product

§20.5(ii) Logarithmic Derivatives

§20.5(iii) Double Products