DLMF:19.25.E27 (Q6533)

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DLMF:19.25.E27
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    Statements

    Defining formula
    2 ⁢ ( z - x ) - 1 / 2 ⁢ R G ⁡ ( x , y , z ) = E ⁡ ( ϕ , k ) + ( cot ⁡ ϕ ) 2 ⁢ F ⁡ ( ϕ , k ) + ( cot ⁡ ϕ ) ⁢ 1 - k 2 ⁢ sin 2 ⁡ ϕ . 2 superscript 𝑧 𝑥 1 2 Carlson-integral-RG 𝑥 𝑦 𝑧 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript italic-ϕ 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle 2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,% k\right)+(\cot\phi)^{2}F\left(\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin^{2}}% \phi}.}}
    0 references
    DLMF id
    DLMF:19.25.E27
    0 references
    Symbols used
    symmetric elliptic integral of second kind
    Defining formula
    R G ⁡ ( x , y , z ) Carlson-integral-RG 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E3.m2abdec
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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
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    Legendre’s incomplete elliptic integral of the first kind
    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.m2agdec
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    Legendre’s incomplete elliptic integral of the second kind
    Defining formula
    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.m2agdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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