DLMF:24.7.E7 (Q7483): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1afdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
test:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.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.m3afdec
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.m1fdec
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.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12112 / rank
 
Normal rank
Property / Symbols used: Q12112 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12112 / qualifier
 
xml-id: C24.S1.XMD7.m1dec

Latest revision as of 00:34, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:24.7.E7
No description defined

    Statements

    test
    B 2 ⁢ n ⁡ ( x ) = ( - 1 ) n + 1 ⁢ 2 ⁢ n ⁢ ∫ 0 ∞ cos ⁡ ( 2 ⁢ π ⁢ x ) - e - 2 ⁢ π ⁢ t cosh ⁡ ( 2 ⁢ π ⁢ t ) - cos ⁡ ( 2 ⁢ π ⁢ x ) ⁢ t 2 ⁢ n - 1 ⁢ d t , Bernoulli-polynomial-B 2 𝑛 𝑥 superscript 1 𝑛 1 2 𝑛 superscript subscript 0 2 𝜋 𝑥 superscript 𝑒 2 𝜋 𝑡 2 𝜋 𝑡 2 𝜋 𝑥 superscript 𝑡 2 𝑛 1 𝑡 {\displaystyle{\displaystyle B_{2n}\left(x\right)=(-1)^{n+1}2n\*\int_{0}^{% \infty}\frac{\cos\left(2\pi x\right)-e^{-2\pi t}}{\cosh\left(2\pi t\right)-% \cos\left(2\pi x\right)}t^{2n-1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:24.7.E7
    0 references
    constraint
    n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}}
    0 references
    Symbols used
    Q11288
    test
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    Q12110
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1fdec
    0 references
    Q12111
    test
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C24.S1.XMD6.m1fdec
    0 references
    Q12112
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C24.S1.XMD7.m1dec
    0 references
     
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