DLMF:5.8.E3 (Q2087): Difference between revisions

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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.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11281 / rank
 
Normal rank
Property / Symbols used: Q11281 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11281 / qualifier
 
xml-id: C5.S1.XMD5.m1dec
Property / Symbols used
 
Property / Symbols used: Q11278 / rank
 
Normal rank
Property / Symbols used: Q11278 / qualifier
 
test:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11278 / qualifier
 
xml-id: C5.S1.XMD6.m1dec

Latest revision as of 18:34, 1 January 2020

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English
DLMF:5.8.E3
No description defined

    Statements

    test
    | Γ ⁡ ( x ) Γ ⁡ ( x + i ⁢ y ) | 2 = ∏ k = 0 ∞ ( 1 + y 2 ( x + k ) 2 ) , superscript Euler-Gamma 𝑥 Euler-Gamma 𝑥 imaginary-unit 𝑦 2 superscript subscript product 𝑘 0 1 superscript 𝑦 2 superscript 𝑥 𝑘 2 {\displaystyle{\displaystyle\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+% \mathrm{i}y\right)}\right|^{2}=\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^% {2}}\right),}}
    0 references
    DLMF id
    DLMF:5.8.E3
    0 references
    constraint
    x ≠ 0 , - 1 , … 𝑥 0 1 … {\displaystyle{\displaystyle x\neq 0,-1,\dots}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1bdec
    0 references
    Q11281
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C5.S1.XMD5.m1dec
    0 references
    Q11278
    test
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C5.S1.XMD6.m1dec
    0 references
     
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