20 Theta Functions Properties 20.4 Values at

§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
§20.5(ii) Logarithmic Derivatives
§20.5(iii) Double Products

§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 $\theta_{1}\left(z,q\right)=2q^{1/4}\sin z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1-2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer, : complex and : 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 $\theta_{2}\left(z,q\right)=2q^{1/4}\cos z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1+2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, : integer, : complex and : 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 $\theta_{3}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right),$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, : integer, : complex and : 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 $\theta_{4}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right).$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, : integer, : complex and : 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 $\theta_{1}\left(z\middle|\tau\right)=\theta_{1}'\left(0\middle|\tau\right)\sin z% \prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z\right)\sin\left(n\pi\tau-z% \right)}{{\sin}^{2}\left(n\pi\tau\right)},$

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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, : integer, : 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 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\cos z% \prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z\right)\cos\left(n\pi\tau-z% \right)}{{\cos}^{2}\left(n\pi\tau\right)},$

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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, : integer, : 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 $\theta_{3}\left(z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})\pi\tau+z\right)\cos\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\cos}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)},$

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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, : integer, : 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 $\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})\pi\tau+z\right)\sin\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\sin}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)}.$

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

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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, : integer, : complex, $\tau$ : lattice parameter and : 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 $\frac{\theta_{1}'\left(z,q\right)}{\theta_{1}\left(z,q\right)}-\cot z=4\sin% \left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos\left(2z\right)+q% ^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin\left(2nz\right),$

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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, : integer, : complex and : 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 $\frac{\theta_{2}'\left(z,q\right)}{\theta_{2}\left(z,q\right)}+\tan z=-4\sin% \left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos\left(2z\right)+q% ^{4n}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin\left(2nz\right).$

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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, : integer, : complex and : 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 $\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left(z,q\right)}=-4\sin\left(2z% \right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos\left(2z\right)+q^{4n% -2}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right),$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer, : complex and : 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 $\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left(z,q\right)}=4\sin\left(2z% \right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos\left(2z\right)+q^{4n% -2}}=4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\sin\left(2nz\right).$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : integer, : complex and : 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 -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 $\theta_{1}\left(z\middle|\tau\right)=z\theta_{1}'\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{\begin{subarray}{c}m=% -M\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),$

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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, : integer, : integer, : 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 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+% \frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right),$

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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, : integer, : integer, : 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 $\theta_{3}\left(z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+% \frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}\right),$

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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, : integer, : integer, : 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 $\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\*% \lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+% \frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right).$

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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, : integer, : integer, : 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

= 0 20.6 Power Series

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§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