DLMF:19.2.E16 (Q6112)

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DLMF:19.2.E16
No description defined

    Statements

    test
    el3 ⁡ ( x , k c , p ) = ∫ 0 arctan ⁡ x d θ ( cos 2 ⁡ θ + p ⁢ sin 2 ⁡ θ ) ⁢ cos 2 ⁡ θ + k c 2 ⁢ sin 2 ⁡ θ = Π ⁡ ( arctan ⁡ x , 1 - p , k ) , Bulirsch-integral-el-3 𝑥 subscript 𝑘 𝑐 𝑝 superscript subscript 0 𝑥 𝜃 2 𝜃 𝑝 2 𝜃 2 𝜃 superscript subscript 𝑘 𝑐 2 2 𝜃 elliptic-integral-third-kind-Pi 𝑥 1 𝑝 𝑘 {\displaystyle{\displaystyle\mathrm{el3}\left(x,k_{c},p\right)=\int_{0}^{% \operatorname{arctan}x}\frac{\mathrm{d}\theta}{({\cos^{2}}\theta+p{\sin^{2}}% \theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{\sin^{2}}\theta}}=\Pi\left(% \operatorname{arctan}x,1-p,k\right),}}
    0 references
    DLMF id
    DLMF:19.2.E16
    0 references
    constraint
    x 2 ≠ - 1 / p superscript 𝑥 2 1 𝑝 {\displaystyle{\displaystyle x^{2}\neq-1/p}}
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    Symbols used
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    Legendre’s incomplete elliptic integral of the third kind
    test
    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.m2abdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10816
    test
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7aadec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
    0 references
    Q11812
    test
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C19.S2.XMD6.m1bdec
    0 references
    Q11813
    test
    k c subscript 𝑘 𝑐 {\displaystyle{\displaystyle k_{c}}}
    xml-id
    C19.S2.XMD7.m1ddec
    0 references
    Q11814
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C19.S2.XMD8.m1bdec
    0 references
     
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