20 Theta Functions Properties 20.4 Values at

§20.5 Infinite Products and Related Results

Keywords:: infinite products, theta functions
Referenced by:: §17.3(iv)
Permalink:: http://dlmf.nist.gov/20.5

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

§20.5(i) Single Products

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

20.5.1 $\mathop{\theta_{{1}}\/}\nolimits\!\left(z,q\right)=2q^{{1/4}}\mathop{\sin\/}% \nolimits z\prod\limits_{{n=1}}^{{\infty}}{\left(1-q^{{2n}}\right)}{\left(1-2q% ^{{2n}}\mathop{\cos\/}\nolimits\!\left(2z\right)+q^{{4n}}\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

20.5.2 $\mathop{\theta_{{2}}\/}\nolimits\!\left(z,q\right)=2q^{{1/4}}\mathop{\cos\/}% \nolimits z\prod\limits_{{n=1}}^{{\infty}}{\left(1-q^{{2n}}\right)}{\left(1+2q% ^{{2n}}\mathop{\cos\/}\nolimits\!\left(2z\right)+q^{{4n}}\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.5.E2
Encodings:: TeX, pMML, png

20.5.3 $\mathop{\theta_{{3}}\/}\nolimits\!\left(z,q\right)=\prod\limits_{{n=1}}^{{% \infty}}\left(1-q^{{2n}}\right)\left(1+2q^{{2n-1}}\mathop{\cos\/}\nolimits\!% \left(2z\right)+q^{{4n-2}}\right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : 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

20.5.4 $\mathop{\theta_{{4}}\/}\nolimits\!\left(z,q\right)=\prod\limits_{{n=1}}^{{% \infty}}\left(1-q^{{2n}}\right)\left(1-2q^{{2n-1}}\mathop{\cos\/}\nolimits\!% \left(2z\right)+q^{{4n-2}}\right).$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : 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

20.5.5 $\mathop{\theta_{{1}}\/}\nolimits\!\left(z\middle|\tau\right)={\mathop{\theta_{% {1}}\/}\nolimits^{{\prime}}}\!\left(0\middle|\tau\right)\mathop{\sin\/}% \nolimits z\prod_{{n=1}}^{{\infty}}\frac{\mathop{\sin\/}\nolimits\!\left(n\pi% \tau+z\right)\mathop{\sin\/}\nolimits\!\left(n\pi\tau-z\right)}{{\mathop{\sin% \/}\nolimits^{{2}}}\!\left(n\pi\tau\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\sin\/}\nolimits z$ : 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

20.5.6 $\mathop{\theta_{{2}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 2}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\cos\/}\nolimits z\prod_{{n% =1}}^{{\infty}}\frac{\mathop{\cos\/}\nolimits\!\left(n\pi\tau+z\right)\mathop{% \cos\/}\nolimits\!\left(n\pi\tau-z\right)}{{\mathop{\cos\/}\nolimits^{{2}}}\!% \left(n\pi\tau\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, : integer, : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E6
Encodings:: TeX, pMML, png

20.5.7 $\mathop{\theta_{{3}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 3}}\/}\nolimits\!\left(0\middle|\tau\right)\prod_{{n=1}}^{{\infty}}\frac{% \mathop{\cos\/}\nolimits\!\left((n-\tfrac{1}{2})\pi\tau+z\right)\mathop{\cos\/% }\nolimits\!\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\mathop{\cos\/}\nolimits^% {{2}}}\!\left((n-\tfrac{1}{2})\pi\tau\right)},$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, : integer, : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E7
Encodings:: TeX, pMML, png

20.5.8 $\mathop{\theta_{{4}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 4}}\/}\nolimits\!\left(0\middle|\tau\right)\prod_{{n=1}}^{{\infty}}\frac{% \mathop{\sin\/}\nolimits\!\left((n-\tfrac{1}{2})\pi\tau+z\right)\mathop{\sin\/% }\nolimits\!\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\mathop{\sin\/}\nolimits^% {{2}}}\!\left((n-\tfrac{1}{2})\pi\tau\right)}.$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\sin\/}\nolimits z$ : 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

¶ Jacobi’s Triple Product

Keywords:: Jacobi’s triple product, theta functions

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : 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

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

§20.5(ii) Logarithmic Derivatives

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

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

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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

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

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex and : nome
A&S Ref:: 16.29.3
Permalink:: http://dlmf.nist.gov/20.5.E12
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

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).
Keywords:: infinite products, theta functions
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/20.5.iii

20.5.14 $\mathop{\theta_{{1}}\/}\nolimits\!\left(z\middle|\tau\right)=z{\mathop{\theta_% {{1}}\/}\nolimits^{{\prime}}}\!\left(0\middle|\tau\right)\*\lim_{{N\to\infty}}% \prod_{{n=-N}}^{{N}}\lim_{{M\to\infty}}\prod_{{\substack{m=-M\\ \left|m\right|+\left|n\right|\neq 0}}}^{{M}}\left(1+\frac{z}{(m+n\tau)\pi}% \right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : integer, : integer, : complex and $\tau$ : lattice parameter
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.5.E14
Encodings:: TeX, pMML, png

20.5.15 $\mathop{\theta_{{2}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 2}}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim_{{N\to\infty}}\prod_{{n=-N}}% ^{{N}}\lim_{{M\to\infty}}\prod_{{m=1-M}}^{{M}}\left(1+\frac{z}{(m-\tfrac{1}{2}% +n\tau)\pi}\right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : integer, : integer, : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E15
Encodings:: TeX, pMML, png

20.5.16 $\mathop{\theta_{{3}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 3}}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim_{{N\to\infty}}\prod_{{n=1-N}% }^{{N}}\lim_{{M\to\infty}}\prod_{{m=1-M}}^{{M}}\left(1+\frac{z}{(m-\tfrac{1}{2% }+(n-\tfrac{1}{2})\tau)\pi}\right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : integer, : integer, : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.5.E16
Encodings:: TeX, pMML, png

20.5.17 $\mathop{\theta_{{4}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta_{{% 4}}\/}\nolimits\!\left(0\middle|\tau\right)\*\lim_{{N\to\infty}}\prod_{{n=1-N}% }^{{N}}\lim_{{M\to\infty}}\prod_{{m=-M}}^{{M}}\left(1+\frac{z}{(m+(n-\tfrac{1}% {2})\tau)\pi}\right).$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : integer, : integer, : complex and $\tau$ : lattice parameter
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.5.E17
Encodings:: TeX, pMML, png

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)).

= 0 20.6 Power Series

§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