DLMF:5.13.E5 (Q2162): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}

\realpart@@{(a_{k})}>0
Property / constraint: ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}

\Re(a_{k})>0
Property / constraint: ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}} / rank
 
Normal rank
Property / constraint
 

k = 1 , 2 , 3 , 4 𝑘 1 2 3 4 {\displaystyle{\displaystyle k=1,2,3,4}}

k=1,2,3,4
Property / constraint: k = 1 , 2 , 3 , 4 𝑘 1 2 3 4 {\displaystyle{\displaystyle k=1,2,3,4}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2addec
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
 
Defining formula:

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.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

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: 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.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3addec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11292 / rank
 
Normal rank
Property / Symbols used: Q11292 / qualifier
 
Defining formula:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q11292 / qualifier
 
xml-id: C5.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q11275 / rank
 
Normal rank
Property / Symbols used: Q11275 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11275 / qualifier
 
xml-id: C5.S1.XMD8.m1ddec

Latest revision as of 01:05, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:5.13.E5
No description defined

    Statements

    Defining formula
    1 4 ⁢ π ⁢ ∫ - ∞ ∞ ∏ k = 1 4 Γ ⁡ ( a k + i ⁢ t ) ⁢ Γ ⁡ ( a k - i ⁢ t ) Γ ⁡ ( 2 ⁢ i ⁢ t ) ⁢ Γ ⁡ ( - 2 ⁢ i ⁢ t ) ⁢ d t = ∏ 1 ≤ j < k ≤ 4 Γ ⁡ ( a j + a k ) Γ ⁡ ( a 1 + a 2 + a 3 + a 4 ) , 1 4 𝜋 superscript subscript superscript subscript product 𝑘 1 4 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 𝑡 subscript product 1 𝑗 𝑘 4 Euler-Gamma subscript 𝑎 𝑗 subscript 𝑎 𝑘 Euler-Gamma subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑎 4 {\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{% k=1}^{4}\Gamma\left(a_{k}+it\right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2% it\right)\Gamma\left(-2it\right)}\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}% \Gamma\left(a_{j}+a_{k}\right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},}}
    0 references
    DLMF id
    DLMF:5.13.E5
    0 references
    constraint
    ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}
    0 references
    ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}
    0 references
    k = 1 , 2 , 3 , 4 𝑘 1 2 3 4 {\displaystyle{\displaystyle k=1,2,3,4}}
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q11292
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C5.S1.XMD1.m1dec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1dec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1ddec
    0 references
     
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