DLMF:2.10.E22 (Q1009): Difference between revisions

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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.m1addec
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.m2agdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5addec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2ahdec
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.m3addec
Property / Symbols used
 
Property / Symbols used: Q11126 / rank
 
Normal rank
Property / Symbols used: Q11126 / qualifier
 
test:

𝒞 𝒞 {\displaystyle{\displaystyle\mathscr{C}}}

\mathscr{C}
Property / Symbols used: Q11126 / qualifier
 
xml-id: C2.S10.XMD12.m1bdec

Latest revision as of 16:36, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:2.10.E22
No description defined

    Statements

    test
    ∑ j = 0 n - 1 x j ( j ! ) 3 = ∫ - 1 / 2 n - ( 1 / 2 ) x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t - ∫ 𝒞 1 x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t e - 2 ⁢ π ⁢ i ⁢ t - 1 + ∫ 𝒞 2 x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t e 2 ⁢ π ⁢ i ⁢ t - 1 , superscript subscript 𝑗 0 𝑛 1 superscript 𝑥 𝑗 superscript 𝑗 3 superscript subscript 1 2 𝑛 1 2 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 subscript subscript 𝒞 1 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 superscript 𝑒 2 𝜋 𝑖 𝑡 1 subscript subscript 𝒞 2 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 superscript 𝑒 2 𝜋 𝑖 𝑡 1 {\displaystyle{\displaystyle\sum_{j=0}^{n-1}\frac{x^{j}}{(j!)^{3}}=\int_{-1/2}% ^{n-(1/2)}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\mathrm{d}t-\int_{\mathscr% {C}_{1}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{\mathrm{d}t}{e^{-2\pi it% }-1}+\int_{\mathscr{C}_{2}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{% \mathrm{d}t}{e^{2\pi it}-1},}}
    0 references
    DLMF id
    DLMF:2.10.E22
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q11126
    test
    𝒞 𝒞 {\displaystyle{\displaystyle\mathscr{C}}}
    xml-id
    C2.S10.XMD12.m1bdec
    0 references
     
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