DLMF:19.2.E4 (Q6085)

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DLMF:19.2.E4
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    Statements

    Defining formula
    F ⁡ ( ϕ , k ) = ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ = ∫ 0 sin ⁡ ϕ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 , elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 superscript subscript 0 italic-ϕ 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 {\displaystyle{\displaystyle F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\mathrm% {d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{\mathrm{d}% t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}},}}
    0 references
    DLMF id
    DLMF:19.2.E4
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
     
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