DLMF:19.25.E11 (Q6515): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / rank
 
Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / qualifier
 
test:

E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}

\incellintEk@{\NVar{\phi}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / qualifier
 
xml-id: C19.S2.E5.m2addec

Revision as of 13:06, 2 January 2020

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DLMF:19.25.E11
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    Statements

    test
    E ⁡ ( ϕ , k ) = - 1 3 ⁢ k ′ 2 ⁢ R D ⁡ ( c - k 2 , c , c - 1 ) + ( c - k 2 ) / ( c ⁢ ( c - 1 ) ) , elliptic-integral-second-kind-E italic-ϕ 𝑘 1 3 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 𝑐 superscript 𝑘 2 𝑐 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 1 {\displaystyle{\displaystyle E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2% }R_{D}\left(c-k^{2},c,c-1\right)+\sqrt{(c-k^{2})/(c(c-1))},}}
    0 references
    DLMF id
    DLMF:19.25.E11
    0 references
    constraint
    ϕ ≠ 1 2 ⁢ π italic-ϕ 1 2 𝜋 {\displaystyle{\displaystyle\phi\neq\tfrac{1}{2}\pi}}
    0 references
    Symbols used
    elliptic integral symmetric in only two variables
    test
    R D ⁡ ( x , y , z ) Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Legendre’s incomplete elliptic integral of the second kind
    test
    E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E5.m2addec
    0 references
     
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