DLMF:19.25.E26 (Q7228)

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DLMF:19.25.E26
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    Statements

    DLMF defining formula
    ( z - x ) 3 / 2 ⁢ R J ⁡ ( x , y , z , p ) = ( 3 / α 2 ) ⁢ ( Π ⁡ ( ϕ , α 2 , k ) - F ⁡ ( ϕ , k ) ) , superscript 𝑧 𝑥 3 2 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 3 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}% ){(\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right))},}}
    0 references
    Symbols used
    symmetric elliptic integral of third kind
    DLMF defining formula
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2afdec
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    Legendre’s incomplete elliptic integral of the first kind
    DLMF defining formula
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2afdec
    0 references
    Legendre’s incomplete elliptic integral of the third kind
    DLMF defining formula
    Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}
    xml-id
    C19.S2.E7.m2acdec
    0 references
    real or complex argument
    DLMF defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1pdec
    0 references
    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1ydec
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    real or complex parameter
    DLMF defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1gdec
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    Identifiers

     
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