DLMF:19.2.E7 (Q6088)

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DLMF:19.2.E7
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    Statements

    test
    Π ⁡ ( ϕ , α 2 , k ) = ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ ( 1 - α 2 ⁢ sin 2 ⁡ θ ) = ∫ 0 sin ⁡ ϕ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 ⁢ ( 1 - α 2 ⁢ t 2 ) . elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 1 superscript 𝛼 2 2 𝜃 superscript subscript 0 italic-ϕ 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 1 superscript 𝛼 2 superscript 𝑡 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}% \frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}% \theta)}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2% }}(1-\alpha^{2}t^{2})}.}}
    0 references
    DLMF id
    DLMF:19.2.E7
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    Symbols used
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
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    Q11807
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1cdec
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    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
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    Q11809
    test
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1dec
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