DLMF:23.15.E9 (Q7339)

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DLMF:23.15.E9
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    Statements

    Defining formula
    η ⁡ ( τ ) = ( 1 2 ⁢ θ 1 ′ ⁡ ( 0 , q ) ) 1 / 3 = e i ⁢ π ⁢ τ / 12 ⁢ θ 3 ⁡ ( 1 2 ⁢ π ⁢ ( 1 + τ ) | 3 ⁢ τ ) . Dedekind-modular-Eta 𝜏 superscript 1 2 diffop Jacobi-theta 1 1 0 𝑞 1 3 superscript 𝑒 𝑖 𝜋 𝜏 12 Jacobi-theta-tau 3 1 2 𝜋 1 𝜏 3 𝜏 {\displaystyle{\displaystyle\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}% '\left(0,q\right)\right)^{1/3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(% 1+\tau)\middle|3\tau\right).}}
    0 references
    DLMF id
    DLMF:23.15.E9
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    Symbols used
    Dedekind’s eta function (or Dedekind modular function)
    Defining formula
    η ⁡ ( τ ) Dedekind-modular-Eta 𝜏 {\displaystyle{\displaystyle\eta\left(\NVar{\tau}\right)}}
    xml-id
    C27.S14.E12.m2adec
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    Q11945
    Defining formula
    θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}
    xml-id
    C20.S2.SS1.m2addec
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    Q11946
    Defining formula
    θ j ⁡ ( z | τ ) Jacobi-theta-tau 𝑗 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}% \right)}}
    xml-id
    C20.S2.SS1.m1adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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