§20.4 Values at $z$ = 0

Keywords:: derivatives, theta functions
Permalink:: http://dlmf.nist.gov/20.4

§20.4(i) Functions and First Derivatives
§20.4(ii) Higher Derivatives

§20.4(i) Functions and First Derivatives

Notes:: See Walker (1996, pp. 90–92), Whittaker and Watson (1927, pp. 470–471), and Lawden (1989, pp. 12–15). Note that these are special cases of (20.5.1)–(20.5.4).
Permalink:: http://dlmf.nist.gov/20.4.i

20.4.1 $\mathop{\theta _{{1}}\/}\nolimits\!\left(0,q\right)={\mathop{\theta _{{2}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)={\mathop{\theta _{{3}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)=0,$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E1
Encodings:: TeX, pMML, png

20.4.2 ${\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)=2q^{{1/4}}\prod\limits _{{n=1}}^{{\infty}}\left(1-q^{{2n}}\right)^{3},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E2
Encodings:: TeX, pMML, png

20.4.3 $\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)=2q^{{1/4}}\prod\limits _{{n=1}}^{{\infty}}\left(1-q^{{2n}}\right)\left(1+q^{{2n}}\right)^{2},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E3
Encodings:: TeX, pMML, png

20.4.4 $\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)=\prod\limits _{{n=1}}^{{\infty}}\left(1-q^{{2n}}\right)\left(1+q^{{2n-1}}\right)^{2},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E4
Encodings:: TeX, pMML, png

20.4.5 $\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)=\prod\limits _{{n=1}}^{{\infty}}\left(1-q^{{2n}}\right)\left(1-q^{{2n-1}}\right)^{2}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E5
Encodings:: TeX, pMML, png

¶ Jacobi’s Identity

Keywords:: Jacobi’s triple product, theta functions

20.4.6 ${\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)=\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
A&S Ref:: 16.28.6
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.4.E6
Encodings:: TeX, pMML, png

§20.4(ii) Higher Derivatives

Notes:: See Whittaker and Watson (1927, p. 471).
Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.4.ii

20.4.7 ${\mathop{\theta _{{1}}\/}\nolimits^{{\prime\prime}}}(0,q)={\mathop{\theta _{{2}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)={\mathop{\theta _{{3}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)={\mathop{\theta _{{4}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)=0.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E7
Encodings:: TeX, pMML, png

20.4.8 $\frac{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)}{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}=-1+24\sum _{{n=1}}^{{\infty}}\frac{q^{{2n}}}{(1-q^{{2n}})^{2}}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/20.4.E8
Encodings:: TeX, pMML, png

20.4.9 $\frac{{\mathop{\theta _{{2}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)}=-1-8\sum _{{n=1}}^{{\infty}}\frac{q^{{2n}}}{(1+q^{{2n}})^{2}},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E9
Encodings:: TeX, pMML, png

20.4.10 $\frac{{\mathop{\theta _{{3}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)}=-8\sum _{{n=1}}^{{\infty}}\frac{q^{{2n-1}}}{(1+q^{{2n-1}})^{2}},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E10
Encodings:: TeX, pMML, png

20.4.11 $\frac{{\mathop{\theta _{{4}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)}=8\sum _{{n=1}}^{{\infty}}\frac{q^{{2n-1}}}{(1-q^{{2n-1}})^{2}}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E11
Encodings:: TeX, pMML, png

20.4.12 $\frac{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(0,q\right)}{{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}=\frac{{\mathop{\theta _{{2}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)}+\frac{{\mathop{\theta _{{3}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{3}}\/}\nolimits\!\left(0,q\right)}+\frac{{\mathop{\theta _{{4}}\/}\nolimits^{{\prime\prime}}}\!\left(0,q\right)}{\mathop{\theta _{{4}}\/}\nolimits\!\left(0,q\right)}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E12
Encodings:: TeX, pMML, png

§20.4 Values at = 0

Contents

§20.4(i) Functions and First Derivatives

¶ Jacobi’s Identity

§20.4(ii) Higher Derivatives

§20.4 Values at $z$ = 0