DLMF:5.17.E3 (Q2183): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11295 / rank
 
Normal rank
Property / Symbols used: Q11295 / qualifier
 
test:

G ⁡ ( z ) Barnes-Gamma 𝑧 {\displaystyle{\displaystyle G\left(\NVar{z}\right)}}

\BarnesG@{\NVar{z}}
Property / Symbols used: Q11295 / qualifier
 
xml-id: C5.S17.E1.m1abdec
Property / Symbols used
 
Property / Symbols used: Euler’s constant / rank
 
Normal rank
Property / Symbols used: Euler’s constant / qualifier
 
test:

γ {\displaystyle{\displaystyle\gamma}}

\EulerConstant
Property / Symbols used: Euler’s constant / qualifier
 
xml-id: C5.S2.E3.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
test:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2adec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q11273 / rank
 
Normal rank
Property / Symbols used: Q11273 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11273 / qualifier
 
xml-id: C5.S1.XMD7.m1adec

Latest revision as of 00:08, 2 January 2020

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DLMF:5.17.E3
No description defined

    Statements

    test
    G ⁡ ( z + 1 ) = ( 2 ⁢ π ) z / 2 ⁢ exp ⁡ ( - 1 2 ⁢ z ⁢ ( z + 1 ) - 1 2 ⁢ γ ⁢ z 2 ) ⁢ ∏ k = 1 ∞ ( ( 1 + z k ) k ⁢ exp ⁡ ( - z + z 2 2 ⁢ k ) ) . Barnes-Gamma 𝑧 1 superscript 2 𝜋 𝑧 2 1 2 𝑧 𝑧 1 1 2 superscript 𝑧 2 superscript subscript product 𝑘 1 superscript 1 𝑧 𝑘 𝑘 𝑧 superscript 𝑧 2 2 𝑘 {\displaystyle{\displaystyle G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1% }{2}z(z+1)-\tfrac{1}{2}\gamma z^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+% \frac{z}{k}\right)^{k}\exp\left(-z+\frac{z^{2}}{2k}\right)\right).}}
    0 references
    DLMF id
    DLMF:5.17.E3
    0 references
    Symbols used
    Q11295
    test
    G ⁡ ( z ) Barnes-Gamma 𝑧 {\displaystyle{\displaystyle G\left(\NVar{z}\right)}}
    xml-id
    C5.S17.E1.m1abdec
    0 references
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    exponential function
    test
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
    0 references
    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1dec
    0 references
    Q11273
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1adec
    0 references
     
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