DLMF:6.16.E2 (Q2311)

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DLMF:6.16.E2
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    Statements

    Defining formula
    S n ⁢ ( x ) = ∑ k = 0 n - 1 sin ⁡ ( ( 2 ⁢ k + 1 ) ⁢ x ) 2 ⁢ k + 1 = 1 2 ⁢ ∫ 0 x sin ⁡ ( 2 ⁢ n ⁢ t ) sin ⁡ t ⁢ d t = 1 2 ⁢ Si ⁡ ( 2 ⁢ n ⁢ x ) + R n ⁢ ( x ) , subscript 𝑆 𝑛 𝑥 superscript subscript 𝑘 0 𝑛 1 2 𝑘 1 𝑥 2 𝑘 1 1 2 superscript subscript 0 𝑥 2 𝑛 𝑡 𝑡 𝑡 1 2 sine-integral 2 𝑛 𝑥 subscript 𝑅 𝑛 𝑥 {\displaystyle{\displaystyle S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x% \right)}{2k+1}=\frac{1}{2}\int_{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}% \mathrm{d}t=\tfrac{1}{2}\mathrm{Si}\left(2nx\right)+R_{n}(x),}}
    0 references
    DLMF id
    DLMF:6.16.E2
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    sine integral
    Defining formula
    Si ⁡ ( z ) sine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Si}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E9.m2adec
    0 references
     
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