DLMF:19.7.E10 (Q6901)

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DLMF:19.7.E10
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    Statements

    DLMF defining formula
    ( 1 - α 2 ) ⁢ Π ⁡ ( ϕ , α 2 , k ) + ( 1 - ω 2 ) ⁢ Π ⁡ ( ϕ , ω 2 , k ) = F ⁡ ( ϕ , k ) + ( 1 - α 2 - ω 2 ) ⁢ c - k 2 ⁢ R C ⁡ ( c ⁢ ( c - 1 ) , ( c - α 2 ) ⁢ ( c - ω 2 ) ) , 1 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 1 superscript 𝜔 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝜔 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 superscript 𝛼 2 superscript 𝜔 2 𝑐 superscript 𝑘 2 Carlson-integral-RC 𝑐 𝑐 1 𝑐 superscript 𝛼 2 𝑐 superscript 𝜔 2 {\displaystyle{\displaystyle(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1% -\omega^{2})\Pi\left(\phi,\omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{% 2}-\omega^{2})\sqrt{c-k^{2}}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})% \right),}}
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    constraint
    ( k 2 - α 2 ) ⁢ ( k 2 - ω 2 ) = k 2 ⁢ ( k 2 - 1 ) superscript 𝑘 2 superscript 𝛼 2 superscript 𝑘 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝑘 2 1 {\displaystyle{\displaystyle(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1% )}}
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    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    DLMF defining formula
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2abdec
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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.m2aedec
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    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.m2aedec
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    real or complex argument
    DLMF defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1fdec
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    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
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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.m1fdec
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    change of variable
    DLMF defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C19.S7.XMD9.m1bdec
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    Identifiers

     
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