DLMF:1.8.E12 (Q266)

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DLMF:1.8.E12
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    Statements

    Defining formula
    ∫ 0 x ( f ⁢ ( t ) - 1 2 ⁢ a 0 ) ⁢ d t = ∑ n = 1 ∞ a n ⁢ sin ⁡ ( n ⁢ x ) + b n ⁢ ( 1 - cos ⁡ ( n ⁢ x ) ) n , subscript superscript 𝑥 0 𝑓 𝑡 1 2 subscript 𝑎 0 𝑡 subscript superscript 𝑛 1 subscript 𝑎 𝑛 𝑛 𝑥 subscript 𝑏 𝑛 1 𝑛 𝑥 𝑛 {\displaystyle{\displaystyle\int^{x}_{0}(f(t)-\tfrac{1}{2}a_{0})\mathrm{d}t=% \sum^{\infty}_{n=1}\frac{a_{n}\sin\left(nx\right)+b_{n}(1-\cos\left(nx\right))% }{n},}}
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    DLMF id
    DLMF:1.8.E12
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    constraint
    0 ≤ x ≤ 2 ⁢ π 0 𝑥 2 𝜋 {\displaystyle{\displaystyle 0\leq x\leq 2\pi}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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