Digital Library of Mathematical Functions
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20 Theta FunctionsProperties

§20.4 Values at z = 0

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§20.4(i) Functions and First Derivatives

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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).
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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,
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function and q: nome
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http://dlmf.nist.gov/20.4.E1
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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},
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
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http://dlmf.nist.gov/20.4.E2
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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},
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
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http://dlmf.nist.gov/20.4.E3
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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},
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
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http://dlmf.nist.gov/20.4.E4
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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}.
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
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Jacobi’s Identity

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).
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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)
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http://dlmf.nist.gov/20.4.E6
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§20.4(ii) Higher Derivatives

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Notes:
See Whittaker and Watson (1927, p. 471).
Keywords:
theta functions
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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.
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function and q: nome
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http://dlmf.nist.gov/20.4.E7
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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+2% 4\sum_{{n=1}}^{{\infty}}\frac{q^{{2n}}}{(1-q^{{2n}})^{2}}.
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
Referenced by:
§23.12
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http://dlmf.nist.gov/20.4.E8
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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}},
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function, n: integer and q: nome
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http://dlmf.nist.gov/20.4.E9
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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}},
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}}.
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% )}.
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Symbols:
\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right): theta function and q: nome
Permalink:
http://dlmf.nist.gov/20.4.E12
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