§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, $n$ : 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

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, $n$ : integer, $z$ : complex and $q$ : 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, $n$ : 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

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, $n$ : 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

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, $n$ : 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

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, $n$ : integer, $z$ : 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, $n$ : integer, $z$ : 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, $n$ : 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

¶ 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, $n$ : 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

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, $n$ : 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

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, $n$ : 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

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, $n$ : integer, $z$ : complex and $q$ : nome
A&S Ref:: 16.29.3
Permalink:: http://dlmf.nist.gov/20.5.E12
Encodings:: TeX, pMML, png

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, $n$ : 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

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

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, $m$ : 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

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, $m$ : integer, $n$ : integer, $z$ : 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, $m$ : integer, $n$ : integer, $z$ : 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, $m$ : 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

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)).

§20.5 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