DLMF:19.8.E9 (Q6217): Difference between revisions

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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.m4aadec
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.m1fdec
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: Q11815 / rank
 
Normal rank
Property / Symbols used: Q11815 / qualifier
 
test:

k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11815 / qualifier
 
xml-id: C19.S1.XMD6.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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11827 / rank
 
Normal rank
Property / Symbols used: Q11827 / qualifier
 
test:

Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle Q_{n}}}

Q_{n}
Property / Symbols used: Q11827 / qualifier
 
xml-id: C19.S8.XMD3.m1bdec

Latest revision as of 12:43, 2 January 2020

No description defined
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English
DLMF:19.8.E9
No description defined

    Statements

    test
    Π ⁡ ( α 2 , k ) = π 4 ⁢ M ⁡ ( 1 , k ′ ) ⁢ k 2 k 2 - α 2 ⁢ ∑ n = 0 ∞ Q n , complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝜋 4 arithmetic-geometric-mean 1 superscript 𝑘 ′ superscript 𝑘 2 superscript 𝑘 2 superscript 𝛼 2 superscript subscript 𝑛 0 subscript 𝑄 𝑛 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,% k^{\prime}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n},}}
    0 references
    DLMF id
    DLMF:19.8.E9
    0 references
    constraint
    1 < α 2 < ∞ 1 superscript 𝛼 2 {\displaystyle{\displaystyle 1<\alpha^{2}<\infty}}
    0 references
    Symbols used
    Q11826
    test
    M ⁡ ( a , g ) arithmetic-geometric-mean 𝑎 𝑔 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
    xml-id
    C19.S8.SS1.p1.m7addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    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.m4aadec
    0 references
    Q11820
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1fdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
    0 references
    Q11815
    test
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1cdec
    0 references
    Q11809
    test
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1adec
    0 references
    Q11827
    test
    Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle Q_{n}}}
    xml-id
    C19.S8.XMD3.m1bdec
    0 references
     
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