DLMF:2.10.E1 (Q988): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11110 / rank
 
Normal rank
Property / Symbols used: Q11110 / qualifier
 
Defining formula:

B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}

\perBernoulliB{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11110 / qualifier
 
xml-id: C24.S2.SS3.m1adec

Revision as of 17:32, 1 January 2020

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DLMF:2.10.E1
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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 ) + ∑ s = 1 m - 1 B 2 ⁢ s ( 2 ⁢ s ) ! ⁢ ( f ( 2 ⁢ s - 1 ) ⁢ ( n ) - f ( 2 ⁢ s - 1 ) ⁢ ( a ) ) + ∫ a n B 2 ⁢ m - B ~ 2 ⁢ m ⁡ ( x ) ( 2 ⁢ m ) ! ⁢ f ( 2 ⁢ m ) ⁢ ( x ) ⁢ d x . superscript subscript 𝑗 𝑎 𝑛 𝑓 𝑗 superscript subscript 𝑎 𝑛 𝑓 𝑥 𝑥 1 2 𝑓 𝑎 1 2 𝑓 𝑛 superscript subscript 𝑠 1 𝑚 1 Bernoulli-number-B 2 𝑠 2 𝑠 superscript 𝑓 2 𝑠 1 𝑛 superscript 𝑓 2 𝑠 1 𝑎 superscript subscript 𝑎 𝑛 Bernoulli-number-B 2 𝑚 periodic-Bernoulli-polynomial-B 2 𝑚 𝑥 2 𝑚 superscript 𝑓 2 𝑚 𝑥 𝑥 {\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)+\sum_{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^% {(2s-1)}(n)-f^{(2s-1)}(a)\right)+\int_{a}^{n}\frac{B_{2m}-\widetilde{B}_{2m}% \left(x\right)}{(2m)!}f^{(2m)}(x)\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:2.10.E1
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11110
    Defining formula
    B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS3.m1adec
    0 references
     
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