DLMF:19.16.E9 (Q6324): Difference between revisions

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Property / constraint
 

z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 ℂ 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,0]}}

z_{j}\in\mathbb{C}\setminus(-\infty,0]
Property / constraint: z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 ℂ 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10923 / rank
 
Normal rank
Property / Symbols used: Q10923 / qualifier
 
Defining formula:

ℂ {\displaystyle{\displaystyle\mathbb{C}}}

\Complexes
Property / Symbols used: Q10923 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r1.m2adec
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.m1aedec
Property / Symbols used
 
Property / Symbols used: Q10784 / rank
 
Normal rank
Property / Symbols used: Q10784 / qualifier
 
Defining formula:

∈ {\displaystyle{\displaystyle\in}}

\in
Property / Symbols used: Q10784 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r10.m2adec
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.m3aedec
Property / Symbols used
 
Property / Symbols used: Q11244 / rank
 
Normal rank
Property / Symbols used: Q11244 / qualifier
 
Defining formula:

( a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b}]}}

(\NVar{a},\NVar{b}]
Property / Symbols used: Q11244 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r1.m3adec
Property / Symbols used
 
Property / Symbols used: Q10812 / rank
 
Normal rank
Property / Symbols used: Q10812 / qualifier
 
Defining formula:

ℝ {\displaystyle{\displaystyle\mathbb{R}}}

\Reals
Property / Symbols used: Q10812 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r14.m2adec
Property / Symbols used
 
Property / Symbols used: Q10785 / rank
 
Normal rank
Property / Symbols used: Q10785 / qualifier
 
Defining formula:

∖ {\displaystyle{\displaystyle\setminus}}

\setminus
Property / Symbols used: Q10785 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r19.m2adec
Property / Symbols used
 
Property / Symbols used: Q11820 / rank
 
Normal rank
Property / Symbols used: Q11820 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11820 / qualifier
 
xml-id: C19.S1.XMD3.m1adec

Latest revision as of 13:52, 2 January 2020

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English
DLMF:19.16.E9
No description defined

    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = 1 B ⁡ ( a , a ′ ) ⁢ ∫ 0 ∞ t a ′ - 1 ⁢ ∏ j = 1 n ( t + z j ) - b j ⁢ d t = 1 B ⁡ ( a , a ′ ) ⁢ ∫ 0 ∞ t a - 1 ⁢ ∏ j = 1 n ( 1 + t ⁢ z j ) - b j ⁢ d t , Carlson-integral-R 𝑎 𝐛 𝐳 1 Euler-Beta 𝑎 superscript 𝑎 ′ superscript subscript 0 superscript 𝑡 superscript 𝑎 ′ 1 subscript superscript product 𝑛 𝑗 1 superscript 𝑡 subscript 𝑧 𝑗 subscript 𝑏 𝑗 𝑡 1 Euler-Beta 𝑎 superscript 𝑎 ′ superscript subscript 0 superscript 𝑡 𝑎 1 subscript superscript product 𝑛 𝑗 1 superscript 1 𝑡 subscript 𝑧 𝑗 subscript 𝑏 𝑗 𝑡 {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}% {\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n% }_{j=1}(t+z_{j})^{-b_{j}}\mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a-1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:19.16.E9
    0 references
    constraint
    b j ∈ ℝ subscript 𝑏 𝑗 {\displaystyle{\displaystyle b_{j}\in\mathbb{R}}}
    0 references
    z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,0]}}
    0 references
    b 1 + ⋯ + b n > a > 0 subscript 𝑏 1 ⋯ subscript 𝑏 𝑛 𝑎 0 {\displaystyle{\displaystyle b_{1}+\cdots+b_{n}>a>0}}
    0 references
    b j ∈ ℝ subscript 𝑏 𝑗 ℝ {\displaystyle{\displaystyle b_{j}\in\mathbb{R}}}
    0 references
    z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 ℂ 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,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.m2adec
    0 references
    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q11244
    Defining formula
    ( a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p2.t1.r1.m3adec
    0 references
    Q10812
    Defining formula
    ℝ {\displaystyle{\displaystyle\mathbb{R}}}
    xml-id
    introduction.Sx4.p2.t1.r14.m2adec
    0 references
    Q10785
    Defining formula
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2adec
    0 references
    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1adec
    0 references
     
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