DLMF:19.23.E8 (Q6479): Difference between revisions

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Property / constraint
 

ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}}

\realpart@@{z_{1}},\realpart@@{z_{2}}>0
Property / constraint: ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}} / rank
 
Normal rank
Property / constraint
 

b 1 , b 2 > 0 subscript 𝑏 1 subscript 𝑏 2 0 {\displaystyle{\displaystyle b_{1},b_{2}>0}}

b_{1},b_{2}>0
Property / constraint: b 1 , b 2 > 0 subscript 𝑏 1 subscript 𝑏 2 0 {\displaystyle{\displaystyle b_{1},b_{2}>0}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}}

\Re z_{1},\Re z_{2}>0
Property / constraint: ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: multivariate hypergeometric function / rank
 
Normal rank
Property / Symbols used: multivariate hypergeometric function / qualifier
 
Defining formula:

R - a ⁡ ( b 1 , … , b n ; z 1 , … , z n ) Carlson-integral-R 𝑎 subscript 𝑏 1 … subscript 𝑏 𝑛 subscript 𝑧 1 … subscript 𝑧 𝑛 {\displaystyle{\displaystyle R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}% };\NVar{z_{1}},\dots,\NVar{z_{n}}\right)}}

\Carlsonmultivarhyper{\NVar{-a}}@{\NVar{b_{1}},\dots,\NVar{b_{n}}}{\NVar{z_{1}},\dots,\NVar{z_{n}}}
Property / Symbols used: multivariate hypergeometric function / qualifier
 
xml-id: C19.S16.E9.m2adec
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: 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.m2afdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2afdec
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.m1agdec
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.m3agdec
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: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aedec

Latest revision as of 14:03, 2 January 2020

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English
DLMF:19.23.E8
No description defined

    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = 2 B ⁡ ( b 1 , b 2 ) ⁢ ∫ 0 π / 2 ( z 1 ⁢ cos 2 ⁡ θ + z 2 ⁢ sin 2 ⁡ θ ) - a ⁢ ( cos ⁡ θ ) 2 ⁢ b 1 - 1 ⁢ ( sin ⁡ θ ) 2 ⁢ b 2 - 1 ⁢ d θ , Carlson-integral-R 𝑎 𝐛 𝐳 2 Euler-Beta subscript 𝑏 1 subscript 𝑏 2 superscript subscript 0 𝜋 2 superscript subscript 𝑧 1 2 𝜃 subscript 𝑧 2 2 𝜃 𝑎 superscript 𝜃 2 subscript 𝑏 1 1 superscript 𝜃 2 subscript 𝑏 2 1 𝜃 {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{2}% {\mathrm{B}\left(b_{1},b_{2}\right)}\int_{0}^{\pi/2}{(z_{1}{\cos^{2}}\theta+z_% {2}{\sin^{2}}\theta)}^{-a}\*(\cos\theta)^{2b_{1}-1}(\sin\theta)^{2b_{2}-1}% \mathrm{d}\theta,}}
    0 references
    DLMF id
    DLMF:19.23.E8
    0 references
    constraint
    ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}}
    0 references
    b 1 , b 2 > 0 subscript 𝑏 1 subscript 𝑏 2 0 {\displaystyle{\displaystyle b_{1},b_{2}>0}}
    0 references
    ℜ ⁡ z 1 , ℜ ⁡ z 2 > 0 subscript 𝑧 1 subscript 𝑧 2 0 {\displaystyle{\displaystyle\Re z_{1},\Re z_{2}>0}}
    0 references
    Symbols used
    multivariate hypergeometric function
    Defining formula
    R - a ⁡ ( b 1 , … , b n ; z 1 , … , z n ) Carlson-integral-R 𝑎 subscript 𝑏 1 … subscript 𝑏 𝑛 subscript 𝑧 1 … subscript 𝑧 𝑛 {\displaystyle{\displaystyle R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}% };\NVar{z_{1}},\dots,\NVar{z_{n}}\right)}}
    xml-id
    C19.S16.E9.m2adec
    0 references
    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
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
     
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