DLMF:20.5.E12 (Q6779)

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DLMF:20.5.E12
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    Statements

    Defining formula
    θ 3 ′ ⁡ ( z , q ) θ 3 ⁡ ( z , q ) = - 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n - 1 1 + 2 ⁢ q 2 ⁢ n - 1 ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n - 2 = 4 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) , diffop Jacobi-theta 3 1 𝑧 𝑞 Jacobi-theta 3 𝑧 𝑞 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 4 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{\theta_{3}'\left(z,q\right)}{\theta_{3}\left% (z,q\right)}=-4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-% 1}\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{% 2n}}\sin\left(2nz\right),}}
    0 references
    DLMF id
    DLMF:20.5.E12
    0 references
    Symbols used
    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.m2afdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ahdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1kdec
    0 references
    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1kdec
    0 references
    Q11950
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C20.S1.XMD5.m1gdec
    0 references
     
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