DLMF:24.8.E9 (Q7496): Difference between revisions

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Property / constraint
 

n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}}

n=1,2,\dots
Property / constraint: n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q11238 / rank
 
Normal rank
Property / Symbols used: Q11238 / qualifier
 
test:

E n Euler-number-E 𝑛 {\displaystyle{\displaystyle E_{\NVar{n}}}}

\EulernumberE{\NVar{n}}
Property / Symbols used: Q11238 / qualifier
 
xml-id: C24.S2.SS2.m1adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2agdec
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.m2addec
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: Q12113 / rank
 
Normal rank
Property / Symbols used: Q12113 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12113 / qualifier
 
xml-id: C24.S1.XMD2.m1hdec
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.m1hdec

Latest revision as of 00:38, 2 January 2020

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DLMF:24.8.E9
No description defined

    Statements

    test
    E 2 ⁢ n = ( - 1 ) n ⁢ ∑ k = 1 ∞ k 2 ⁢ n cosh ⁡ ( 1 2 ⁢ π ⁢ k ) - 4 ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ( 2 ⁢ k + 1 ) 2 ⁢ n e 2 ⁢ π ⁢ ( 2 ⁢ k + 1 ) - 1 , Euler-number-E 2 𝑛 superscript 1 𝑛 superscript subscript 𝑘 1 superscript 𝑘 2 𝑛 1 2 𝜋 𝑘 4 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 2 𝑘 1 2 𝑛 superscript 𝑒 2 𝜋 2 𝑘 1 1 {\displaystyle{\displaystyle{}E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{% \cosh\left(\tfrac{1}{2}\pi k\right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^% {2n}}{e^{2\pi(2k+1)}-1},}}
    0 references
    DLMF id
    DLMF:24.8.E9
    0 references
    constraint
    n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}}
    0 references
    Symbols used
    Q11238
    test
    E n Euler-number-E 𝑛 {\displaystyle{\displaystyle E_{\NVar{n}}}}
    xml-id
    C24.S2.SS2.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    Q12113
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C24.S1.XMD2.m1hdec
    0 references
    Q12110
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1hdec
    0 references
     
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