DLMF:2.10.E2 (Q989)

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DLMF:2.10.E2
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    Statements

    Defining formula
    ∑ j = a n f ⁢ ( j ) = ∫ a n f ⁢ ( x ) ⁢ d x + 1 2 ⁢ f ⁢ ( a ) + 1 2 ⁢ f ⁢ ( n ) - 2 ⁢ ∫ 0 ∞ ℑ ⁡ ( f ⁢ ( a + i ⁢ y ) ) e 2 ⁢ π ⁢ y - 1 ⁢ d y + ∑ s = 1 m B 2 ⁢ s ( 2 ⁢ s ) ! ⁢ f ( 2 ⁢ s - 1 ) ⁢ ( n ) + 2 ⁢ ( - 1 ) m ( 2 ⁢ m ) ! ⁢ ∫ 0 ∞ ℑ ⁡ ( f ( 2 ⁢ m ) ⁢ ( n + i ⁢ ϑ n ⁢ y ) ) ⁢ y 2 ⁢ m ⁢ d y e 2 ⁢ π ⁢ y - 1 , superscript subscript 𝑗 𝑎 𝑛 𝑓 𝑗 superscript subscript 𝑎 𝑛 𝑓 𝑥 𝑥 1 2 𝑓 𝑎 1 2 𝑓 𝑛 2 superscript subscript 0 𝑓 𝑎 𝑖 𝑦 superscript 𝑒 2 𝜋 𝑦 1 𝑦 superscript subscript 𝑠 1 𝑚 Bernoulli-number-B 2 𝑠 2 𝑠 superscript 𝑓 2 𝑠 1 𝑛 2 superscript 1 𝑚 2 𝑚 superscript subscript 0 superscript 𝑓 2 𝑚 𝑛 𝑖 subscript italic-ϑ 𝑛 𝑦 superscript 𝑦 2 𝑚 𝑦 superscript 𝑒 2 𝜋 𝑦 1 {\displaystyle{\displaystyle\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\mathrm{d}x+% \tfrac{1}{2}f(a)+\tfrac{1}{2}f(n)-2\int_{0}^{\infty}\frac{\Im(f(a+iy))}{e^{2% \pi y}-1}\mathrm{d}y+\sum_{s=1}^{m}\frac{B_{2s}}{(2s)!}f^{(2s-1)}(n)+2\frac{(-% 1)^{m}}{(2m)!}\int_{0}^{\infty}\Im(f^{(2m)}(n+i\vartheta_{n}y))\frac{y^{2m}% \mathrm{d}y}{e^{2\pi y}-1},}}
    0 references
    DLMF id
    DLMF:2.10.E2
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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