DLMF:24.13.E9 (Q7543): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12114 / rank
 
Normal rank
Property / Symbols used: Q12114 / qualifier
 
Defining formula:

E n ⁡ ( x ) Euler-polynomial-E 𝑛 𝑥 {\displaystyle{\displaystyle E_{\NVar{n}}\left(\NVar{x}\right)}}

\EulerpolyE{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q12114 / qualifier
 
xml-id: C24.S2.SS2.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1ahdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3ahdec
Property / Symbols used
 
Property / Symbols used: Q12110 / rank
 
Normal rank
Property / Symbols used: Q12110 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12110 / qualifier
 
xml-id: C24.S1.XMD5.m1hdec
Property / Symbols used
 
Property / Symbols used: Q12111 / rank
 
Normal rank
Property / Symbols used: Q12111 / qualifier
 
Defining formula:

t 𝑡 {\displaystyle{\displaystyle t}}

t
Property / Symbols used: Q12111 / qualifier
 
xml-id: C24.S1.XMD6.m1hdec

Latest revision as of 01:44, 2 January 2020

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DLMF:24.13.E9
No description defined

    Statements

    Defining formula
    ∫ 0 1 / 2 E 2 ⁢ n ⁡ ( t ) ⁢ d t = - E 2 ⁢ n + 1 ⁡ ( 0 ) 2 ⁢ n + 1 = 2 ⁢ ( 2 2 ⁢ n + 2 - 1 ) ⁢ B 2 ⁢ n + 2 ( 2 ⁢ n + 1 ) ⁢ ( 2 ⁢ n + 2 ) , superscript subscript 0 1 2 Euler-polynomial-E 2 𝑛 𝑡 𝑡 Euler-polynomial-E 2 𝑛 1 0 2 𝑛 1 2 superscript 2 2 𝑛 2 1 Bernoulli-number-B 2 𝑛 2 2 𝑛 1 2 𝑛 2 {\displaystyle{\displaystyle\int_{0}^{1/2}E_{2n}\left(t\right)\mathrm{d}t=-% \frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)}% ,}}
    0 references
    DLMF id
    DLMF:24.13.E9
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1acdec
    0 references
    Q12114
    Defining formula
    E n ⁡ ( x ) Euler-polynomial-E 𝑛 𝑥 {\displaystyle{\displaystyle E_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS2.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q12110
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1hdec
    0 references
    Q12111
    Defining formula
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C24.S1.XMD6.m1hdec
    0 references
     
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