DLMF:24.7.E2 (Q7478): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosecant function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosecant function / qualifier
 
test:

csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}

\csch@@{\NVar{z}}
Property / Symbols used: hyperbolic cosecant function / qualifier
 
xml-id: C4.S28.E5.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aadec
Property / Symbols used
 
Property / Symbols used: Q12110 / rank
 
Normal rank
Property / Symbols used: Q12110 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12110 / qualifier
 
xml-id: C24.S1.XMD5.m1adec
Property / Symbols used
 
Property / Symbols used: Q12111 / rank
 
Normal rank
Property / Symbols used: Q12111 / qualifier
 
test:

t 𝑡 {\displaystyle{\displaystyle t}}

t
Property / Symbols used: Q12111 / qualifier
 
xml-id: C24.S1.XMD6.m1adec

Latest revision as of 01:33, 2 January 2020

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DLMF:24.7.E2
No description defined

    Statements

    test
    B 2 ⁢ n = ( - 1 ) n + 1 ⁢ 4 ⁢ n ⁢ ∫ 0 ∞ t 2 ⁢ n - 1 e 2 ⁢ π ⁢ t - 1 ⁢ d t = ( - 1 ) n + 1 ⁢ 2 ⁢ n ⁢ ∫ 0 ∞ t 2 ⁢ n - 1 ⁢ e - π ⁢ t ⁢ csch ⁡ ( π ⁢ t ) ⁢ d t , Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 4 𝑛 superscript subscript 0 superscript 𝑡 2 𝑛 1 superscript 𝑒 2 𝜋 𝑡 1 𝑡 superscript 1 𝑛 1 2 𝑛 superscript subscript 0 superscript 𝑡 2 𝑛 1 superscript 𝑒 𝜋 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1% }}{e^{2\pi t}-1}\mathrm{d}t=(-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}% \operatorname{csch}\left(\pi t\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:24.7.E2
    0 references
    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    hyperbolic cosecant function
    test
    csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}
    xml-id
    C4.S28.E5.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q12110
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1adec
    0 references
    Q12111
    test
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C24.S1.XMD6.m1adec
    0 references
     
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