DLMF:19.5.E4 (Q6133): 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
 
test:

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: Q11816 / rank
 
Normal rank
Property / Symbols used: Q11816 / qualifier
 
test:

Π ⁡ ( α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}

\compellintPik@{\NVar{\alpha}^{2}}{\NVar{k}}
Property / Symbols used: Q11816 / qualifier
 
xml-id: C19.S2.E8.m4adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec
Property / Symbols used
 
Property / Symbols used: Q11818 / rank
 
Normal rank
Property / Symbols used: Q11818 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11818 / qualifier
 
xml-id: C19.S1.XMD2.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11820 / rank
 
Normal rank
Property / Symbols used: Q11820 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11820 / qualifier
 
xml-id: C19.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11809 / rank
 
Normal rank
Property / Symbols used: Q11809 / qualifier
 
test:

α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}

\alpha^{2}
Property / Symbols used: Q11809 / qualifier
 
xml-id: C19.S1.XMD7.m1dec

Latest revision as of 12:39, 2 January 2020

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English
DLMF:19.5.E4
No description defined

    Statements

    test
    Π ⁡ ( α 2 , k ) = π 2 ⁢ ∑ n = 0 ∞ ( 1 2 ) n n ! ⁢ ∑ m = 0 n ( 1 2 ) m m ! ⁢ k 2 ⁢ m ⁢ α 2 ⁢ n - 2 ⁢ m = π 2 ⁢ F 1 ⁡ ( 1 2 ; 1 2 , 1 ; 1 ; k 2 , α 2 ) , complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝜋 2 superscript subscript 𝑛 0 Pochhammer 1 2 𝑛 𝑛 superscript subscript 𝑚 0 𝑛 Pochhammer 1 2 𝑚 𝑚 superscript 𝑘 2 𝑚 superscript 𝛼 2 𝑛 2 𝑚 𝜋 2 Appell-F-1 1 2 1 2 1 1 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=% 0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{% \left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}% \left(\tfrac{1}{2};\tfrac{1}{2},1;1;k^{2},\alpha^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.5.E4
    0 references
    Symbols used
    first Appell function
    test
    F 1 ⁡ ( α ; β , β ′ ; γ ; missing , missing ) Appell-F-1 𝛼 𝛽 superscript 𝛽 ′ 𝛾 missing missing {\displaystyle{\displaystyle{F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{% \beta^{\prime}};\NVar{\gamma};missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S13.E1.m2adec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q11816
    test
    Π ⁡ ( α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m4adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    Q11818
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C19.S1.XMD2.m1cdec
    0 references
    Q11820
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1dec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
    0 references
    Q11809
    test
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1dec
    0 references
     
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