DLMF:5.12.E9 (Q2154): Difference between revisions

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Property / constraint
 

0 < c < 1 0 𝑐 1 {\displaystyle{\displaystyle 0<c<1}}

0<c<1
Property / constraint: 0 < c < 1 0 𝑐 1 {\displaystyle{\displaystyle 0<c<1}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}

\Re(a+b)>0
Property / constraint: ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: beta function / rank
 
Normal rank
Property / Symbols used: beta function / qualifier
 
Defining formula:

B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}

\EulerBeta@{\NVar{a}}{\NVar{b}}
Property / Symbols used: beta function / qualifier
 
xml-id: C5.S12.E1.m2ahdec
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.m2aedec
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.m1ahdec
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.m3ahdec
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.m1addec
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.m1hdec
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.m1hdec

Latest revision as of 19:45, 1 January 2020

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DLMF:5.12.E9
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ c - ∞ ⁢ i c + ∞ ⁢ i t - a ⁢ ( 1 - t ) - 1 - b ⁢ d t = 1 b ⁢ B ⁡ ( a , b ) , 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑡 𝑎 superscript 1 𝑡 1 𝑏 𝑡 1 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle{\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{% -a}(1-t)^{-1-b}\mathrm{d}t=\frac{1}{b\mathrm{B}\left(a,b\right)}},}}
    0 references
    DLMF id
    DLMF:5.12.E9
    0 references
    constraint
    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
    0 references
    0 < c < 1 0 𝑐 1 {\displaystyle{\displaystyle 0<c<1}}
    0 references
    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
    0 references
    Symbols used
    beta function
    Defining formula
    B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C5.S12.E1.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    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.m3ahdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1hdec
    0 references
    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1hdec
    0 references
     
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