DLMF:5.13.E3 (Q2160): Difference between revisions

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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.m2abdec
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.m1abdec
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.m2abdec
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.m3abdec
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.m1aadec
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.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11283 / rank
 
Normal rank
Property / Symbols used: Q11283 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11283 / qualifier
 
xml-id: C5.S1.XMD9.m1bdec

Latest revision as of 01:04, 2 January 2020

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DLMF:5.13.E3
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ Γ ⁡ ( a + i ⁢ t ) ⁢ Γ ⁡ ( b + i ⁢ t ) ⁢ Γ ⁡ ( c - i ⁢ t ) ⁢ Γ ⁡ ( d - i ⁢ t ) ⁢ d t = Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( a + d ) ⁢ Γ ⁡ ( b + c ) ⁢ Γ ⁡ ( b + d ) Γ ⁡ ( a + b + c + d ) , 1 2 𝜋 superscript subscript Euler-Gamma 𝑎 𝑖 𝑡 Euler-Gamma 𝑏 𝑖 𝑡 Euler-Gamma 𝑐 𝑖 𝑡 Euler-Gamma 𝑑 𝑖 𝑡 𝑡 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑐 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a% +it\right)\Gamma\left(b+it\right)\Gamma\left(c-it\right)\Gamma\left(d-it\right% )\mathrm{d}t=\frac{\Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c% \right)\Gamma\left(b+d\right)}{\Gamma\left(a+b+c+d\right)},}}
    0 references
    DLMF id
    DLMF:5.13.E3
    0 references
    constraint
    ℜ ⁡ a , ℜ ⁡ b , ℜ ⁡ c , ℜ ⁡ d > 0 𝑎 𝑏 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re a,\Re b,\Re c,\Re d>0}}
    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.m2abdec
    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
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1bdec
    0 references
    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1bdec
    0 references
     
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