DLMF:19.8.E6 (Q6212): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11600 / rank
 
Normal rank
Property / Symbols used: Q11600 / qualifier
 
test:

K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}

\compellintKk@{\NVar{k}}
Property / Symbols used: Q11600 / qualifier
 
xml-id: C19.S2.E8.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11601 / rank
 
Normal rank
Property / Symbols used: Q11601 / qualifier
 
test:

E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}

\compellintEk@{\NVar{k}}
Property / Symbols used: Q11601 / qualifier
 
xml-id: C19.S2.E8.m2adec
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.m1cdec
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.m1adec
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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11824 / rank
 
Normal rank
Property / Symbols used: Q11824 / qualifier
 
test:

a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}

a_{n}
Property / Symbols used: Q11824 / qualifier
 
xml-id: C19.S8.XMD1.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11825 / rank
 
Normal rank
Property / Symbols used: Q11825 / qualifier
 
test:

g n subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n}}}

g_{n}
Property / Symbols used: Q11825 / qualifier
 
xml-id: C19.S8.XMD2.m1ddec

Latest revision as of 12:42, 2 January 2020

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DLMF:19.8.E6
No description defined

    Statements

    test
    E ⁡ ( k ) = π 2 ⁢ M ⁡ ( 1 , k ′ ) ⁢ ( a 0 2 - ∑ n = 0 ∞ 2 n - 1 ⁢ c n 2 ) = K ⁡ ( k ) ⁢ ( a 1 2 - ∑ n = 2 ∞ 2 n - 1 ⁢ c n 2 ) , complete-elliptic-integral-second-kind-E 𝑘 𝜋 2 arithmetic-geometric-mean 1 superscript 𝑘 ′ superscript subscript 𝑎 0 2 superscript subscript 𝑛 0 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 complete-elliptic-integral-first-kind-K 𝑘 superscript subscript 𝑎 1 2 superscript subscript 𝑛 2 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 {\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}% \right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k% \right)\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.8.E6
    0 references
    constraint
    g 0 = k ′ subscript 𝑔 0 superscript 𝑘 ′ {\displaystyle{\displaystyle g_{0}=k^{\prime}}}
    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.m7abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11600
    test
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aadec
    0 references
    Q11601
    test
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2adec
    0 references
    Q11820
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1cdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1adec
    0 references
    Q11815
    test
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1adec
    0 references
    Q11824
    test
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C19.S8.XMD1.m1ddec
    0 references
    Q11825
    test
    g n subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n}}}
    xml-id
    C19.S8.XMD2.m1ddec
    0 references
     
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