DLMF:20.2.E11 (Q6751)

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DLMF:20.2.E11
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    Statements

    Defining formula
    θ 1 ⁡ ( z | τ ) = - θ 2 ⁡ ( z + 1 2 ⁢ π | τ ) = - i ⁢ M ⁢ θ 4 ⁡ ( z + 1 2 ⁢ π ⁢ τ | τ ) = - i ⁢ M ⁢ θ 3 ⁡ ( z + 1 2 ⁢ π + 1 2 ⁢ π ⁢ τ | τ ) , Jacobi-theta-tau 1 𝑧 𝜏 Jacobi-theta-tau 2 𝑧 1 2 𝜋 𝜏 𝑖 𝑀 Jacobi-theta-tau 4 𝑧 1 2 𝜋 𝜏 𝜏 𝑖 𝑀 Jacobi-theta-tau 3 𝑧 1 2 𝜋 1 2 𝜋 𝜏 𝜏 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=-\theta_{2}% \left(z+\tfrac{1}{2}\pi\middle|\tau\right)=-iM\theta_{4}\left(z+\tfrac{1}{2}% \pi\tau\middle|\tau\right)=-iM\theta_{3}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}% \pi\tau\middle|\tau\right),}}
    0 references
    DLMF id
    DLMF:20.2.E11
    0 references
    Symbols used
    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.m1aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1kdec
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    Q11949
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C20.S1.XMD4.m1kdec
    0 references
    DLMF:20.2.E10
    Defining formula
    M 𝑀 {\displaystyle{\displaystyle M}}
    xml-id
    C20.S2.XMD1.m1adec
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