DLMF:23.8.E2 (Q7284)

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DLMF:23.8.E2
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    Statements

    Defining formula
    ζ ⁡ ( z ) - η 1 ⁢ z ω 1 - π 2 ⁢ ω 1 ⁢ cot ⁡ ( π ⁢ z 2 ⁢ ω 1 ) = 2 ⁢ π ω 1 ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - q 2 ⁢ n ⁢ sin ⁡ ( n ⁢ π ⁢ z ω 1 ) . Weierstrass-zeta-on-lattice 𝑧 𝕃 subscript 𝜂 1 𝑧 subscript 𝜔 1 𝜋 2 subscript 𝜔 1 𝜋 𝑧 2 subscript 𝜔 1 2 𝜋 subscript 𝜔 1 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 𝑛 𝜋 𝑧 subscript 𝜔 1 {\displaystyle{\displaystyle\zeta\left(z\right)-\frac{\eta_{1}z}{\omega_{1}}-% \frac{\pi}{2\omega_{1}}\cot\left(\frac{\pi z}{2\omega_{1}}\right)=\frac{2\pi}{% \omega_{1}}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin\left(\frac{n\pi z}{% \omega_{1}}\right).}}
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    DLMF id
    DLMF:23.8.E2
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    Symbols used
    DLMF:23.2.E5
    Defining formula
    ζ ⁡ ( z ) Weierstrass-zeta-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\zeta\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E5.m2adec
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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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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1adec
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    Q12066
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C23.S1.XMD10.m1adec
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    Q12060
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C23.S1.XMD3.m1adec
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    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1adec
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    Q12049
    Defining formula
    ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}
    xml-id
    C23.S1.XMD8.m1adec
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    Q12059
    Defining formula
    η j subscript 𝜂 𝑗 {\displaystyle{\displaystyle\eta_{j}}}
    xml-id
    C23.S2.XMD6.m1adec
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