DLMF:2.10.E20 (Q1007): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2afdec

Revision as of 17:35, 1 January 2020

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DLMF:2.10.E20
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    Statements

    Defining formula
    ∑ j = 0 n - 1 x j ( j ! ) 3 = 1 2 ⁢ i ⁢ ∫ 𝒞 x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ cot ⁡ ( π ⁢ t ) ⁢ d t , superscript subscript 𝑗 0 𝑛 1 superscript 𝑥 𝑗 superscript 𝑗 3 1 2 𝑖 subscript 𝒞 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle\sum_{j=0}^{n-1}\frac{x^{j}}{(j!)^{3}}=\frac{1}{2i% }\int_{\mathscr{C}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\cot\left(\pi t% \right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:2.10.E20
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
     
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