DLMF:19.4.E8 (Q6128)

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DLMF:19.4.E8
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    Statements

    Defining formula
    ( k ⁢ k ′ 2 ⁢ D k 2 + ( 1 - 3 ⁢ k 2 ) ⁢ D k - k ) ⁢ F ⁡ ( ϕ , k ) = - k ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ ( 1 - k 2 ⁢ sin 2 ⁡ ϕ ) 3 / 2 , 𝑘 superscript superscript 𝑘 ′ 2 superscript subscript 𝐷 𝑘 2 1 3 superscript 𝑘 2 subscript 𝐷 𝑘 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 italic-ϕ italic-ϕ superscript 1 superscript 𝑘 2 2 italic-ϕ 3 2 {\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F% \left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}},}}
    0 references
    DLMF id
    DLMF:19.4.E8
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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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.m2abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1cdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1gdec
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    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1fdec
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    Q11817
    Defining formula
    D k subscript 𝐷 𝑘 {\displaystyle{\displaystyle D_{k}}}
    xml-id
    C19.S4.XMD1.m1dec
    0 references
     
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