DLMF:20.5.E10 (Q6777): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11950 / rank
 
Normal rank
Property / Symbols used: Q11950 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11950 / qualifier
 
xml-id: C20.S1.XMD5.m1edec

Latest revision as of 14:34, 2 January 2020

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DLMF:20.5.E10
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    Statements

    Defining formula
    θ 1 ′ ⁡ ( z , q ) θ 1 ⁡ ( z , q ) - cot ⁡ z = 4 ⁢ sin ⁡ ( 2 ⁢ z ) ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - 2 ⁢ q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ z ) + q 4 ⁢ n = 4 ⁢ ∑ n = 1 ∞ q 2 ⁢ n 1 - q 2 ⁢ n ⁢ sin ⁡ ( 2 ⁢ n ⁢ z ) , diffop Jacobi-theta 1 1 𝑧 𝑞 Jacobi-theta 1 𝑧 𝑞 𝑧 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 2 superscript 𝑞 2 𝑛 2 𝑧 superscript 𝑞 4 𝑛 4 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{\theta_{1}'\left(z,q\right)}{\theta_{1}\left% (z,q\right)}-\cot z=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^% {2n}\cos\left(2z\right)+q^{4n}}=4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}% \sin\left(2nz\right),}}
    0 references
    DLMF id
    DLMF:20.5.E10
    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.m2addec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1idec
    0 references
    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1idec
    0 references
    Q11950
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C20.S1.XMD5.m1edec
    0 references
     
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