DLMF:25.12.E9 (Q7728)

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DLMF:25.12.E9
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    Statements

    Defining formula
    ∑ n = 1 ∞ sin ⁡ ( n ⁢ θ ) n 2 = - ∫ 0 θ ln ⁡ ( 2 ⁢ sin ⁡ ( 1 2 ⁢ x ) ) ⁢ d x . superscript subscript 𝑛 1 𝑛 𝜃 superscript 𝑛 2 superscript subscript 0 𝜃 2 1 2 𝑥 𝑥 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}% {n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)% \mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:25.12.E9
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1cdec
    0 references
    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1ddec
    0 references
    Q12167
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C25.S12.XMD1.m1bdec
    0 references
     
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